text:
\section{Introduction}\label{section:introduction}
All \emph{partially ordered sets (or posets)} considered here will be finite, graded, and
contain a minimum element (denoted by $\hat{0}$). We assume familiarity with poset and poset
topology terminology and notation. For background on posets the reader should visit
\cite[Chapter 3]{Stanley2012} and \cite{Wachs2007}.

Throughout the paper, $P$ will denote a finite graded poset with a  $\hat{0}$ and $\rho$ will
denote its rank function. The \emph{M\"obius function} of a poset $P$ is defined
recursively for pairs $x<y$ in $P$ by
\begin{equation}\label{equation:mobiusdefinition}
\mu(x,y)=\begin{cases} 1 &\mbox{if } x=y, \\
-\displaystyle \sum_{x\le z < y} \mu(x,z) & \mbox{if } x \neq y. \end{cases}
\end{equation}

We illustrate, with two examples, how to calculate the values $\mu(\hat{0},x)$ of the M\"obius
function. These two examples will be of a particular relevance throughout this article.
\begin{example}\label{example:partitionlattice}
Let $\Pi_n$ denote the poset whose underlying set is formed by the partitions of the set
$[n]:= \{1,2,\dots, n\}$  with order relation given for $\pi,\pi^{\prime} \in \Pi_n$ by $\pi \le
\pi^{\prime}$ if every block of $\pi$ is contained in some block of $\pi^{\prime}$. Equivalently,
the cover relation $\pi \lessdot \pi^{\prime}$ is defined whenever $\pi^{\prime}$ is obtained from
$\pi$ when exactly two blocks of $\pi$ are merged to form a single block in $\pi^{\prime}$ while
the remaining blocks of $\pi$ and $\pi^{\prime}$ are the same. We say that the
partitions are ordered by \emph{refinement} and we call $\Pi_n$ the \emph{partition lattice} (since this poset has additional structure, that of a lattice). In
Figure \ref{figure:examplemobius} we illustrate the values $\mu(\hat{0},x)$ of
the M\"obius function for every $x \in \Pi_3$.
\end{example}

\begin{example}\label{example:ISF}
Let $T$ be a tree with vertices labeled by distinct integers.  We call the smallest vertex of
$T$ the \emph{root}.  We say $T$ is an \emph{increasing tree} if  the sequence of vertex labels
read along any path starting at the root  of $T$ is increasing.  An \emph{increasing spanning
forest} is a
collection of
increasing trees whose vertex labels form a partition of $[n]$.  The word ``spanning"  here
indicates that these forests are spanning forests of the complete graph.  For more information
about increasing
spanning forests
see~\cite{HallamMartinSagan2016}. We use  $\mathcal{ISF}_n$ to denote the set
of increasing spanning forests on $[n]$. A partial
order on
$\mathcal{ISF}_n$ is defined by  $F_1 \lessdot F_2$  if exactly two trees in
$F_1$ are replaced by the tree in $F_2$ that is obtained after joining their roots with an edge.
Note that the root  of the resulting tree is the smaller label among the roots of the
two joined trees. See  Figure \ref{figure:examplemobius} for the Hasse diagram of $\mathcal{ISF}_3$
together with the M\"obius values $\mu(\hat{0},x)$ for every $x \in \mathcal{ISF}_3$.
\end{example}

\begin{figure}
 \input{pictures/example_of_mobius_numbers.tikz}
 \caption{M\"obius function values $\mu(\hat{0},x)$ on $\Pi_3$ and $\mathcal{ISF}_3$}
 \label{figure:examplemobius}
\end{figure}

Two important invariants that we can associate to a graded poset $P$ are its Whitney numbers of the
first and second kind. The \emph{ $k^{th}$ Whitney number of the first kind}, $w_k(P)$, is defined
by
\begin{align}\label{definition:whitneyfirstkind}
w_k(P) = \sum_{\rho(x) =k} \mu(\hat{0},x),
\end{align}
and the \emph{$k^{th}$ Whitney number of the second kind}, $W_k(P)$, is defined by
\begin{align}\label{definition:whitneysecondkind}
W_k(P)  = |\{x\in P \mid \rho(x) = k\}|.
\end{align}

The Whitney numbers of a graded poset play an important role in many areas of combinatorics.  For
example,
they appear as coefficients of the chromatic polynomial of a finite graph~\cite{rota1964}.
Stanley~\cite{Stanley1973} showed they
can be used to count the number of acyclic orientations of a graph.   When the poset is the
intersection lattice of a real hyperplane arrangement, Zaslavsky~\cite{Zaslavsky1975} showed  that
its Whitney numbers can be used to
count the number of bounded and unbounded regions. For complex hyperplane
arrangements the Whitney numbers can be used to compute the dimensions of the
Orlik-Solomon algebra that is isomorphic to the Whitney cohomology of the intersection
lattice~\cite{OrlikSolomon1980}.  Very recently, a long
standing conjecture of Heron~\cite{Heron1972}, Rota~\cite{rota1971} and Welsh~\cite{Welsh1976}
concerning the
log-concavity of the Whitney numbers of the first kind of geometric lattices was settled by
Adiprasito, Huh, and Katz~\cite{AdiprasitoHuhKatz2015} using ideas coming from Hodge theory.

\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|c|c|}\hline
\bf{$k$} & \textbf{$w_k(\Pi_n)$} & \textbf{$W_k(\Pi_n)$} & \textbf{$w_k(\mathcal{ISF}_n)$} &
\textbf{$W_k(\mathcal{ISF}_n)$}\\\hline
\textbf{$0$}  & $1$ & $1$& $1$ & $1$\\\hline
\textbf{$1$}  & $-3$ & $3$& $-3$ & $3$\\\hline
\textbf{$2$}  & $2$ & $1$ & $1$ & $2$\\\hline
\end{tabular}
\end{center}
\caption{Whitney numbers of the first and second kind for $\Pi_3$ for $\mathcal{ISF}_3$.}
\label{table:whitneynumberspi3if3}
\end{table}

Now consider the previous two examples in Figure~\ref{figure:examplemobius}. When we compute the
Whitney
numbers of $\Pi_3$ and $\mathcal{ISF}_3$ and we list them side by side (see Table
\ref{table:whitneynumberspi3if3}), we notice a curious coincidence: (up to sign) their Whitney
numbers of the first and second kind are interchanged. This surprising phenomenon was
initially noticed by the authors of \cite{DleonWachs2016} for a different, but closely related,
pair of posets: the poset $\Pi^w_n$ of
weighted partitions and the poset $\mathcal{SF}_n$ of rooted spanning forests on $[n]$
    studied by Reiner in \cite{Reiner1978} and Sagan in \cite{Sagan1983}. It turns out that this
  phenomenon occurs for many other pairs of posets, in particular, the authors announced in
  \cite{DleonHallam2017} that for every geometric lattice there exist another poset
  with their Whitney numbers interchanged. This seemingly common phenomenon motivates the following
  definition.

  \begin{definition}\label{definition:whitneyduals}
  Let $P$ and $Q$ be graded posets. We say that $P$ and $Q$ are \emph{Whitney Duals} if for all
  $k\ge0$ we have that
  \begin{align}
  |w_k(P)|=W_k(Q) \text{ and } |w_k(Q)|=W_k(P).
  \end{align}
  \end{definition}

  According to this definition, $\Pi_3$ and $\mathcal{ISF}_3$ are Whitney duals. In fact, this is true in general for
  $\Pi_n$ and $\mathcal{ISF}_n$ for $n\ge 1$ (see \cite{DleonHallam2017}).

Our investigation on Whitney duals is driven by the following two questions that we address to
different extents in the present work:

\begin{question}\label{question:whitneydual1}
 When does a graded poset $P$ has a Whitney dual?
\end{question}

\begin{question}\label{question:whitneydual2}
Is there a method to construct a Whitney dual for a graded poset $P$? Perhaps under certain
assumptions about $P$.
\end{question}

In this article we advance towards an answer to the first question as we provide an answer to the
second question.  To do this, we use poset topology tools including edge labeling,  chain-edge labelings, and
quotient posets.

To state our main theorem, we now briefly review a few basic ideas concerning edge and chain-edge labelings. Recall that the \emph{Hasse diagram of $P$} is the directed graph on $P$ with directed
edges the
covering relations $x\lessdot y$ in $P$, i.e., $x < y$ where there is no
$z\in P$ satisfying $x<z<y$.
Also recall  that an \emph{edge labeling or E-labeling} of $P$ is a map $\lambda: \mathcal{E}(P)\rightarrow
\Lambda$ where $\mathcal{E}(P)$ is the set of edges of the Hasse
diagram of $P$ and $\Lambda$ is some other poset of labels. An
edge labeling is said to be an \emph{ER-labeling} if in every interval $[x,y]$ of $P$ there is a
unique \emph{saturated or unrefinable chain} $$\c :
(x
= x_0 \lessdot x_1 \lessdot \cdots \lessdot x_{\ell-1} \lessdot x_{\ell}= y)$$  that is
\emph{increasing}, i.e., such that
$$\lambda(x_0 \lessdot x_1) < \lambda(x_1 \lessdot x_2) < \cdots < \lambda(x_{\ell-1}\lessdot
x_{\ell}).$$
The concept of an  edge labeling was generalized
by Bj\"orner and Wachs to a labeling of pairs
formed by maximal chains and edges of the Hasse diagram known as \emph{chain-edge labelings or C-labelings}. The generalization of an ER-labeling is then known as a  CR-labeling.
Chain-edge
labelings are more involved and technical in their definition than edge labelings. Thus,  as
a presentation strategy across this article we  choose to first discuss  the
definitions and constructions in the context of edge labelings and then explain how these
constructions generalize to the context of chain-edge labelings.
The reader should visit \cite{ Bjorner1980, BjornerWachs1983, Stanley2012, Wachs2007} for information  on edge and edge-chain labelings.

In Section \ref{section:whitneylabelings} we define new types of edge and chain-edge labelings that
we call \emph{Whitney labelings} or \emph{W-labelings}, the name coming from the fact
that they provide sufficient conditions for the construction of Whitney duals of graded posets.  A crucial property of Whitney labelings is the rank two switching property.   We say that an edge labeling $\lambda$ has the
\emph{rank two switching property} if for every maximal chain
$$\c : ( \hat{0}= x_0 \lessdot x_1 \lessdot \cdots \lessdot x_{k-1} \lessdot x_{k})$$ that
has an increasing step $\lambda(x_{i-1} \lessdot x_i) < \lambda(x_i \lessdot x_{i+1})$ at rank $i$
there is a unique maximal chain
$$\c' : ( \hat{0}= x_0 \lessdot x_1 \lessdot \cdots \lessdot x_{i-1} \lessdot x_i' \lessdot
x_{i+1}\lessdot \cdots \lessdot x_{k-1} \lessdot x_{k}),$$
whose labels are the same as the ones from $\c$ except for $\lambda(x_{i-1} \lessdot
x_i')=\lambda(x_{i} \lessdot x_{i+1})$ and $\lambda(x_{i}' \lessdot x_{i+1})=\lambda(x_{i-1}
\lessdot x_i).$
For chain-edge labelings we require an  additional condition for the rank two switching property, namely that the
choice of the $x_i'$ is consistent among maximal chains that coincide in the bottom $d$ elements
when $d>i$.

A Whitney labeling of $P$ is an ER or CR-labeling with the rank two switching
property and with the property that in every interval $[x,y]$ (a rooted interval in
the case of CR-labelings) of $P$ each ascent-free maximal chain is determined uniquely by its
sequence of labels from bottom to top. We will call Whitney labelings respectively EW or CW-labelings depending if the underlying labeling is an ER or a CR-labeling.

The main result of this paper is the following:
\begin{theorem}\label{theorem:WimpliesWhitney}
 Let $P$ be a poset with a Whitney labeling $\lambda$.  Then $P$ has a Whitney dual.
 Moreover, using $\lambda$ we can explicitly construct a Whitney dual $Q_\lambda(P)$ of $P$.
\end{theorem}

We actually prove a more general result. We define a more general kind of labeling that we call a
\emph{generalized Whitney labeling}. Its definition relies on a set of more
general but, at the same time, more technical conditions that also imply the existence of a Whitney dual. However, all the examples that we know of so far satisfy the nicer definition
that we gave above.

Theorem \ref{theorem:WimpliesWhitney} guarantees for every Whitney labeling $\lambda$ of a graded
poset $P$ a Whitney dual $Q_{\lambda}(P)$. In general, there is no reason to expect that this
construction is independent of $\lambda$ and in fact in Section \ref{weightedPartSec} we show
that for the poset of weighted partitions $\Pi_n^w$ there are at least two non-isomorphic Whitney
duals. The two Whitney duals that we find are of the form $Q_{\lambda}(P)$ for two different Whitney labelings.  One of the labelings is an ER-labeling that was already introduced in
\cite{DleonWachs2016}. We
prove here that this labeling is also an EW-labeling. The second labeling is a new CR-labeling that
we also show is a CW-labeling. The ascent-free chains of the new CW-labeling are indexed by
rooted forests of $[n]$ and the construction of the poset $Q_{\lambda}(P)$  provides an explanation
of the Whitney duality between the weighted partition poset, $\Pi_n^w$, and the poset of rooted spanning forest, $\mathcal{SF}_n$. This duality can be seen
from the work of Gonz\'alez D'Le\'on - Wachs in \cite{DleonWachs2016} and the work
of Sagan in \cite{Sagan1983} after comparing the Whitney numbers of $\Pi_n^w$ and $\mathcal{SF}_n$.  Our construction gives an explanation for this duality, which was our initial motivation for the
current project.

\begin{theorem}\label{theorem:weightedpartitionsSF}
For all $n\ge 1$, there is a CW-labeling $\lambda_C$ of $\Pi_n^w$ such that
$Q_{\lambda_C}(\Pi_n^w)$ and $\mathcal{SF}_n$ are isomorphic posets. In particular  $\Pi_n^w$ and  $\mathcal{SF}_n$
are Whitney duals.
\end{theorem}

The rest of the paper is organized as follows.  In Section~\ref{section:basicexamples} we
consider basic notions and examples of Whitney duals.  In Section~\ref{section:whitneylabelings} we
introduce the notion of Whitney labeling and we give a construction of a Whitney dual using a Whitney labeling and quotient posets.

In Section \ref{section:whitneydual} we give a characterization of the M\"obius values of
$Q_\lambda(P)$. We also provide a simpler description of the posets $Q_\lambda(P)$ when $\lambda$ is a
Whitney labeling (in the strict sense of the
definition, i.e., not a generalized one).

In Section \ref{section:examplesWposets} we provide a series of examples of posets for which Theorem \ref{theorem:WimpliesWhitney} applies. These include all geometric lattices, the poset of weighted partitions $\Pi_n^w$ and the
R$^*$S-labelable posets studied by Simion and Stanley \cite{SimionStanley1999}.   This last family
includes the lattice $\mathcal{NC}_n$ of
noncrossing  partitions of $[n]$, the posets of shuffles studied by Greene \cite{Greene1988} and also the noncrossing partition
lattices of types B and D studied by Reiner \cite{Reiner1997} (see also \cite{Hersh1999}). With these
examples we illustrate both the existence of EW and CW-labelings. For the particular
cases of $\mathcal{NC}_n$ and $\Pi_n^w$ we also give explicit combinatorial descriptions of their Whitney
duals $Q_\lambda(P)$.

In Section \ref{section:heckeaction} we show that a Whitney labeling induces a local action
of the $0$-Hecke algebra  $H_n(0)$ of type A on the set of maximal chains of $P$ and hence a
representation $\chi_P$ of $H_n(0)$ on the space spanned by the maximal chains of $P$. This action
can be transported to a local action on the set of maximal chains of its Whitney dual
$Q_\lambda(P)$ and hence also induces  a representation $\chi_{Q_\lambda(P)}$. In particular, we prove the
following theorem.

 \begin{theorem}\label{theorem:goodactions}
Let $P$ be a poset with a (generalized) CW-labeling $\lambda$.  Then
$$
ch(\chi_{Q_{\lambda}(P)})=ch(\chi_P) = F_p(x) = \omega F_{Q_{\lambda}}(x),
$$
where $F_P(x)$ is Ehrenborg's flag quasisymmetric function of the graded poset $P$, $ch(V)$ indicates the quasisymmetric characteristic of the $H_n(0)$-representation $V$ and $\omega$ is
the classical involution in the ring of quasisymmetric functions that maps Gessel's fundamental
quasisymmetric function $L_{S,n}$ indexed by a set $S$ to the one indexed by its complement $[n-1]\setminus S$.
\end{theorem}

Theorem \ref{theorem:goodactions} implies that $\chi_{Q_{\lambda}(P)}$ is induced by a ``good
action'' in the sense of \cite{SimionStanley1999}. The results in \cite{McNamara2003} then
imply that the  bowtie-free  maximal intervals of the posets $Q_{\lambda}(P)$ are \emph{snellable} (see \cite{McNamara2003}) and the maximal intervals of $Q_\lambda(P)$ that are lattices are \emph{supersolvable}.

\section{Basic Examples  of posets with and without Whitney Duals}\label{section:basicexamples}

We start by a discussion on some basic examples of posets which have and do not have Whitney
duals.

A graded poset $P$ is \emph{Eulerian} if $\mu(x,y)  =
(-1)^{\rho(y)-\rho(x)}$ for all $x\leq y$ in $P$ (see Figure~\ref{fig:Eulerian} for an example).
Other examples of Eulerian posets include the face lattice of a convex polytope and the (strong)
Bruhat order.  For more information on Eulerian posets see the survey
article \cite{Stanley1994}.  From the definition of Eulerian poset, it is immediate that we have
$|w_k(P)|
= W_k(P)$ for all $k$.  Therefore, every Eulerian poset has a Whitney dual, namely itself.  It is
natural to ask if all posets which are their own Whitney dual are Eulerian.   The poset in
Figure~\ref{fig:selfdulnoneulerian} shows that this is not the case \footnote{We thank Cyrus Hettle
 for pointing out this example to the authors.}.    This leads to the currently open question of whether there is a natural characterization of self-Whitney dual posets.

\begin{figure}
\begin{subfigure}[b]{0.5\textwidth}
                \centering
 \input{pictures/exampleWhitneyduality_1.tikz}

\caption{Eulerian and self Whitney dual}
                  \label{fig:Eulerian}
               \end{subfigure}~
\begin{subfigure}[b]{0.5\textwidth}
                \centering
\input{pictures/exampleWhitneyduality_2.tikz}

\caption{Self  Whitney dual, but not Eulerian}
                  \label{fig:selfdulnoneulerian}
               \end{subfigure}~

\begin{subfigure}[b]{0.5\textwidth}
                \centering

\input{pictures/exampleWhitneyduality_3.tikz}
\caption{No Whitney duals}
                  \label{fig:posetnowhitneydual}

\end{subfigure}~
\begin{subfigure}[b]{0.5\textwidth}
                \centering

 \input{pictures/exampleWhitneyduality_4.tikz}
\caption{No Whitney duals}
                  \label{fig:posetnowhitneydual2}

\end{subfigure}~

         \caption{Examples of self Whitney-duality and of posets without Whitney dual}
  \label{fig:exampleofselfdualityandnotwhitney}
\end{figure}

Not every ranked poset has a Whitney dual.  For example, consider the three element chain $C$ in
Figure \ref{fig:posetnowhitneydual}.  We have that $w_2(C) =0$ and $W_2(C) =1$.
If $Q$ was a Whitney dual of $C$, then $|w_2(Q)|=1$ and $W_2(Q)=0$, which is clearly impossible.
This illustrates the fact that a poset $P$ with $|w_k(P)|=0$ for some $k$ smaller than the rank of
the poset  cannot
have a Whitney dual.

We also remark that there are more complicated reasons  that can prevent a poset from having a
Whitney dual.  As an example, consider the poset in Figure~\ref{fig:posetnowhitneydual2}.   Suppose
that this poset had a Whitney dual, $Q$.  Then the absolute value of the sequence of Whitney
numbers of the first kind for $Q$ would be $1,3,2,1$.  Moreover, $Q$ would necessarily have a unique maximal element ($\hat{1}$).  This would imply that the sum of the Whitney numbers of the first kind would be 0.
Now
it must be the case that $w_0(Q) = 1$ and $w_2(Q) =-3$.  As a result we should be able to
assign a sign to $2$ and $1$ so that $1-3\pm 2 \pm 1=0$, which is impossible.

One might wonder if Whitney duals are unique.   By considering the posets in
Figures~\ref{fig:Eulerian} and~\ref{fig:selfdulnoneulerian}, one can see this is not the case.
These posets are both self Whitney dual and Whitney dual to each other.  In fact, a poset that is not self-Whitney dual can have multiple non-isomorphic Whitney duals.  We show in
Section~\ref{weightedPartSec} that  $\Pi_n^w$ is an example of such a poset.

Now that we have seen some basic examples of posets with and without Whitney duals, we turn our attention to an approach on constructing Whitney duals.

\section{Whitney labelings and quotient posets}\label{section:whitneylabelings}
The idea of edge labelings or E-labelings is pervasive in the poset literature.
The concept of an ER-labeling (originally called an R-labeling) was introduced by Stanley
(see \cite{Stanley1972}) to study the M\"obius function of rank selected subposets of a graded
poset. Bj\"orner \cite{Bjorner1980} extended this notion by adding a lexicographic condition. Such labelings are called EL-labelings. The extra condition on an EL-labeling implies the shellability of the
order (simplicial) complex of the poset and hence has stronger topological
consequences that allow for the determination of its homotopy type.  Bj\"orner and Wachs
\cite{BjornerWachs1982} extended  the theory of lexicographic shellability to certain type
of
labelings called chain-edge labelings or C-labelings (see also
\cite{BjornerWachs1983,BjornerWachs1996,BjornerWachs1997}). This new concept of a CR-labeling (or CL-labeling when a lexicographic condition is considered) provided a more flexible
description that helped in  the determination of the M\"obius numbers of posets that were not known
to be included in the family of ER-labelable posets.  For example, C-labelings can be used to understand the M\"obius numbers    of the Bruhat order of a
Coxeter group. An ER-labeling is a special case of a
CR-labeling and hence chain-edge labelings are in principle more general than edge labelings. In
fact it is not known whether a poset that has a CR-labeling also has an ER-labeling.
In this section we show how edge and chain-edge labelings can be used to construct Whitney duals.
We will first describe all constructions using only the concept of an E-labeling
since the presentation and proofs will be more clear and pleasant to the reader. The use of
C-labelings require certain technical details, like the
concept of a rooted interval, that might obscure the relevant ideas involved. However, we hope that the reader will find that the bottom-to-top nature of our constructions are compatible with C-labelings.

In the last subsection we discuss how all the theory developed for E-labelings continue to hold
for C-labelings. We describe  the important modifications to the definitions and properties
involved when transferring from the context of E-labelings to C-labelings.

\subsection{Edge labelings}

We now discuss edge labelings and their relation with Whitney numbers.  First, let
us recall some basic facts about edge labelings.  For complete treatments
on the topic, see~\cite{ Bjorner1980, BjornerWachs1983, Stanley2012}.  Let $P$ be a poset, and
let
$\mathcal{E}(P)$ be the set of edges of the Hasse diagram of $P$.  Moreover, let $\Lambda$ be
an arbitrary fixed poset that will be considered as the \emph{poset of labels}.  An \emph{edge
labeling} of $P$ is a map  $\lambda: \mathcal{E}(P)\rightarrow \Lambda$.

Let $P$ be a poset with edge
labeling $\lambda$. To every \emph{saturated chain} (also known as an \emph{unrefinable
chain}) $$\c :
(x
= x_0 \lessdot x_1 \lessdot \cdots \lessdot x_{\ell-1} \lessdot x_{\ell}= y)$$ we can associate a
corresponding \emph{word of labels}
$$\lambda(\c) = \lambda(x_0\lessdot x_1) \lambda(x_1\lessdot x_2) \cdots \lambda(x_{\ell-1}\lessdot
x_{\ell}).$$
We say that  $\c$ is  \emph{increasing} if its word of labels $\lambda(\c)$ is
\emph{strictly}  increasing, that is, $\c$ is  increasing if
\begin{align*}
 \lambda(x_0\lessdot x_1) < \lambda(x_1\lessdot x_2)<  \cdots < \lambda(x_{\ell-1}\lessdot
x_{\ell}).
\end{align*}
  We say that  $\c$ is  \emph{ascent-free} if its word of labels
$\lambda(\c)$ has no ascents, i.e. $\lambda(x_{i-1}\lessdot x_i) \not<  \lambda(x_i\lessdot
x_{i+1}) $,
for
all
$i=1,\dots,\ell-1$.
Clearly, there are chains that are neither increasing nor ascent-free.

\begin{definition}\label{definition:ER}
 An edge labeling is an \emph{ER-labeling} if in each closed
interval $[x,y]$ of $P$, there is a unique  increasing maximal chain. By analogy, we say that an
edge labeling is an
ER$^*$-labeling if
in each closed interval $[x,y]$ of $P$, there is a unique ascent-free maximal chain.
\end{definition}

We note that in~\cite{Stanley2012}, ER-labelings are referred to as R-labelings and  have a
slightly different definition.    First, it is assumed that the labels are
totally ordered. Also,
increasing refers to weakly increasing and ascent-free is replaced with strictly decreasing.
An almost identical proof as the one of Theorem 3.14.2 in~\cite{Stanley2012} gives
the following.

\begin{theorem}[c.f. Theorem 3.14.2 in \cite{Stanley2012}]\label{theorem:muER}
Let $P$ be a graded poset with an ER-labeling (ER$^*$-labeling).  Then
$$
 \mu(x,y)= (-1)^{\rho(y)-\rho(x)}|\{\c \,\mid\,\c \mbox{ is an ascent-free (increasing) maximal
chain
in $[x,y]$}\}|.
$$
\end{theorem}

Let us now consider examples of ER and ER$^*$ labelings.   In both examples the labels will
come from the set $[n]\times [n]$ and we assume that this set is ordered
lexicographically  using  the natural order on $[n]$ as integers.

\begin{example}\label{example:labelingposetofpartitions}
Let $\lambda:\mathcal{E}(\Pi_n)\rightarrow [n]\times [n]$ be the edge
labeling defined on the partition lattice $\Pi_n$ by $\lambda(\pi \lessdot \sigma) = (i,j)$ where
$i<j$ and $i$ and $j$ are the
minimum elements of the two blocks of
$\pi$ that were merged to obtain $\sigma$.  This edge labeling is an ER-labeling and is a special
case of
Bj\"orner's minimum labeling for
geometric lattices described in \cite{Bjorner1982}. In
Figure~\ref{figure:exampleER} the labeling $\lambda$ of $\Pi_3$ is depicted.
\end{example}

\begin{example}\label{example:labelingISF}
Let $\lambda^*:\mathcal{E}(\mathcal{ISF}_n)\rightarrow [n]\times [n]$ be the edge labeling defined on $\mathcal{ISF}_n$ by setting
$\lambda^*(F_1\lessdot F_2)$ to be the unique edge in $F_2$ that is not in $F_1$. It was proved in
\cite[Proposition 3]{DleonHallam2017} that this edge labeling is an ER$^*$-labeling. In
Figure~\ref{figure:exampleER}, the labeling $\lambda^*$ of $\mathcal{ISF}_3$ is
depicted.
\end{example}

\begin{figure}
 \input{pictures/example_of_ER_labelings.tikz}
 \caption{Example of ER and ER$^*$-labelings on $\Pi_3$ and $\mathcal{ISF}_3$}
 \label{figure:exampleER}
\end{figure}

The reader may have noticed how similar are the labelings of $\Pi_3$ and $\mathcal{ISF}_3$.  This
is no coincidence and as we will see later, the labeling of $\mathcal{ISF}_3$ can be obtained from the
labeling of $\Pi_3$.

Using the definition of Whitney numbers, Definition~\ref{definition:ER}, and
Theorem~\ref{theorem:muER}, we can describe the
Whitney numbers of a poset using an ER-labeling (ER$^{*}$-labeling) by the enumeration of
saturated chains as follows.

\begin{proposition}\label{proposition:ERwhitneynumbers}
Let $P$ be a graded poset with an ER-labeling (ER$^{*}$-labeling).  Then $|w_k(P)|$ is the number
of
ascent-free (increasing) saturated chains starting at $\hat{0}$  of length $k$.  Moreover,
$|W_k(P)|$
is the number of increasing (ascent-free) saturated chains starting at $\hat{0}$ of length $k$.
\end{proposition}

In Table \ref{table:propositionERwhitneynumbers} we summarize in the conclusions of Proposition
\ref{proposition:ERwhitneynumbers} to
highlight the importance of this proposition and the fact that ER and ER$^*$-labelings switch the
role of increasing and
ascent-free saturated chains. This fact will be used later in a construction of Whitney duals.

\begin{table}
\begin{center}
\begin{tabular}{|c|c| c| } \hline
$\lambda$ is  an& $|w_k(P)|$ & $W_k(P)$ \\ \hline
 ER-labeling& $\#$ (ascent-free sat. chains& $\#$ (increasing sat. chains \\
  & of length $k$ starting at $\hat{0}$) &of length $k$ starting at $\hat{0}$)\\ \hline
 ER$^*$-labeling&  $\#$ (increasing sat. chains & $\#$ (ascent-free sat. chains\\
 &of length $k$ starting at $\hat{0}$) &of length $k$ starting at $\hat{0}$)\\\hline
\end{tabular}
\end{center}
\caption{Proposition \ref{proposition:ERwhitneynumbers}}
\label{table:propositionERwhitneynumbers}
\end{table}

\subsection{Constructing Whitney Duals}\label{section:constructSec}
In the following we use edge labelings with the following property.

\begin{definition}\label{definition:ranktwoE}
Let $\lambda$ be an edge labeling on $P$. We say that $\lambda$ has the
\emph{rank two switching property} if for every maximal chain
$$\c : ( \hat{0}= x_0 \lessdot x_1 \lessdot \cdots \lessdot x_{k-1} \lessdot x_{k})$$ that
has an increasing step $\lambda(x_{i-1} \lessdot x_i) < \lambda(x_i \lessdot x_{i+1})$ at rank $i$
there is a unique maximal chain
$$\c' : ( \hat{0}= x_0 \lessdot x_1 \lessdot \cdots \lessdot x_{i-1} \lessdot x_i' \lessdot
x_{i+1}\lessdot \cdots \lessdot x_{k-1} \lessdot x_{k}),$$
whose labels are the same as the ones from $\c$ except for $\lambda(x_{i-1} \lessdot
x_i')=\lambda(x_{i} \lessdot x_{i+1})$ and $\lambda(x_{i}' \lessdot x_{i+1})=\lambda(x_{i-1}
\lessdot x_i).$
 \end{definition}
\begin{remark}
For E-labelings, there is a simpler way to describe the rank two switching property. We can say that $\lambda$ has the rank two
switching property  provided that for every interval $[x,y]$ with $\rho(y)-\rho(x)=2$, if $\lambda_1\lambda_2$ is the word of labels of the unique increasing
maximal chain in the interval, then there exists a unique maximal chain in $[x,y]$ whose word of labels is $\lambda_2\lambda_1$.  We choose to give the seemingly more complicated definition because it closely resembles the condition for C-labelings that we provide later.
 \end{remark}

In Figure~\ref{figure:exampleER},  one can see  that the labeling of $\Pi_3$ given in
Example~\ref{example:labelingposetofpartitions} has the rank two switching property.  Indeed,
the increasing chain in the unique rank two interval of $\Pi_3$ is labeled by $(1,2)(1,3)$ and
there is a unique chain labeled by $(1,3)(1,2)$.  In fact, $\Pi_n$ has the rank two switching property for all $n\geq 1$. One can verify
this using the fact that there are only two types of rank two interval in $\Pi_n$. Each interval is
isomorphic to $\Pi_3$ or to a boolean algebra of rank two.

\begin{definition}\label{definition:quadraticexchanges}
Let $P$ be a graded poset and let $\lambda$ be an ER-labeling of $P$ with the rank two switching
property.
Denote $\mathcal{M}_{[x,y]}$ the set of maximal chains in $[x,y]\subseteq P$ and let
$$\c : (x = x_0 \lessdot x_1 \lessdot \cdots \lessdot x_{\ell-1} \lessdot x_{\ell}= y)\in
\mathcal{M}_{[x,y]}$$ be a saturated chain having an ascent $\lambda(x_{i-1}\lessdot x_i)<\lambda(x_i
\lessdot x_{i+1})$ at rank
$i$. By the rank two switching property there is an element $\tilde x_i \in [x_{i-1},x_{i+1}]$ such
that $\lambda(x_{i-1} \lessdot \tilde x_i)=\lambda(x_i \lessdot x_{i+1})$ and
$\lambda(x_{i-1}\lessdot  x_i)= \lambda( \tilde x_i \lessdot x_{i+1})$. We say that the chain $\c
\setminus
\{x_i\} \cup \{\tilde x_i\}$, that is obtained from $\c$ after removing $x_i$ and adding $\tilde
x_i$, was obtained by a \emph{quadratic exchange at rank $i$}.  We will  use the notation
$U_i(\c)=\c'$ if $\c'$ is obtained from $\c$ by applying a quadratic exchange at rank level $i$ and
whenever $\c$ does not have an ascent at rank $i$ we define $U_i(\c)=\c$.
\end{definition}

For every interval $[x,y]$ in $P$ we can define a labeled directed graph $G_{[x,y]}$ whose
vertex
set is $\mathcal{M}_{[x,y]}$ and where there is a labeled directed edge $\c_1
\stackrel{U_i}{\rightharpoonup} \c_2$ if $\c_2$ is obtained from $\c_1$ by a quadratic exchange at
rank $i$.
We define also $S(\c):=\{\lambda(x_{i-1}\lessdot x_i)\,\mid\, i\in [\ell]\}$ the multiset of
labels of the chain $\c$. Note that a quadratic exchange leaves the multiset of labels invariant,
i.e.,
$S(\c_1)=S(\c_2)$ whenever  $\c_1$ and $\c_2$ are related by a sequence of quadratic exchanges.
Thus in general $G_{[x,y]}$ is not a connected graph, for example two chains in $\mathcal{M}_{[x,y]}$ with
different multisets of labels belong to different connected components of $G_{[x,y]}$.

We call a
pair $(i<j)$ such that $\lambda(x_{i-1}\lessdot x_i) < \lambda(x_{j-1}\lessdot x_j)$ a
\emph{label inversion} of $\c$.
Note that a quadratic exchange reduces the
number of label
inversions of $\c$ and, since the directed edges in $G_{[x,y]}$ are given by quadratic exchanges,
this
implies that $G_{[x,y]}$ does not contain directed cycles.

By our construction ascent-free chains in $\mathcal{M}_{[x,y]}$ are precisely the vertices of
$G_{[x,y]}$ that have outdegree $0$, also known as \emph{sinks}. Indeed, quadratic exchanges can
only happen at an ascending step of a saturated chain. In particular, by repeatedly applying
quadratic exchanges we get the following lemma.

\begin{lemma}\label{lemma:switchinglemma}
For every chain $\c \in \mathcal{M}_{[x,y]}$ there exists at
least one ascent-free maximal chain $\c^{\prime}\in \mathcal{M}_{[x,y]}$ such that $\c$ and $\c^{\prime}$
belong to the same connected component of $G_{[x,y]}$. Any such chain $\c^{\prime}$ satisfies
$S(\c^{\prime})=S(\c)$.
\end{lemma}

Lemma \ref{lemma:switchinglemma} tells us that each connected component of $G_{[x,y]}$ has at least
one vertex that is a sink, however this sink vertex is not necessarily unique. Ideally, for our
construction, we would like to have a unique ascent-free chain $\c \in \mathcal{M}_{[x,y]}$ in each connected component of $G_{[x,y]}$. This would imply that the connected components of
$G_{[x,y]}$
are indexed by
ascent-free chains in $\mathcal{M}_{[x,y]}$. To guarantee this condition, we will use a classical result in
graph theory and in the study of term rewriting systems, known either as the Diamond Lemma
or Newman's Lemma (see \cite{Newman1942,Huet1980}).

We say that a directed graph $G$ is \emph{confluent} if for
every pair of vertices $x$ and $y$ in the same connected component of $G$ there are
directed walks $x\rightsquigarrow u$ (a sequence of directed edges $x\rightharpoonup
z \rightharpoonup \cdots
\rightharpoonup u $) and $y\rightsquigarrow u$ that meet at a common vertex
$u$ of $G$. Confluency has the following easy but interesting consequence.

\begin{lemma}\label{lemma:uniquesink}
 If $G$ is a confluent directed graph that does not contain infinite directed walks
$x_0\rightharpoonup x_1\rightharpoonup x_2\rightharpoonup \cdots $ (or cycles) then every
connected
component of $G$ contains a unique sink vertex.
\end{lemma}

We say that $G$ is \emph{locally confluent} if for
every pair of directed edges $v \rightharpoonup x$ and $v \rightharpoonup y$ there are
directed walks $x\rightsquigarrow u$ and $y\rightsquigarrow u$ that meet at a common vertex
$u$ of $G$.

\begin{lemma}[Newman's Lemma c.f. \cite{Newman1942,Huet1980}]\label{lemma:newman}
A directed graph $G$ without infinite directed walks is confluent if and only if it is
locally confluent.
\end{lemma}

Lemma \ref{lemma:newman} simplifies the procedure of checking confluency by restricting to
local confluency which is an easier condition to test, especially given that $G_{[x,y]}$ is a
finite graph
without cycles. In $G_{[x,y]}$ the local confluency condition can be tested by considering two
labeled directed
edges $\c \stackrel{U_i}{\rightharpoonup} \c_1$ and $\c \stackrel{U_j}{\rightharpoonup}\c_2$. Note
that  since $U_i(\c)$ produces a unique chain by the rank two switching property, we may always
assume $i\neq j$.  If $|i-j|>1$ then we can always obtain a chain $\c^{\prime}$ either as $\c
\stackrel{U_i}{\rightharpoonup} \c_1 \stackrel{U_j}{\rightharpoonup} \c^{\prime}$ or $\c
\stackrel{U_j}{\rightharpoonup}\c_2 \stackrel{U_i}{\rightharpoonup} \c^{\prime}$ since the elements
involved in the ascending steps are not adjacent to each other.    In other words, if $|i-j|>1$,
then $U_iU_j(\c) = U_jU_i(\c)$ for all chains $\c$ with ascents at $i$ and $j$.
When $j=i+1$ we say that $\c$ has a
\emph{double-ascent} or that $\c$ has a \emph{critical condition at rank $i$}, i.e., for $\c : (x =
x_0 \lessdot x_1 \lessdot \cdots \lessdot x_{\ell-1} \lessdot x_{\ell}= y)$  we have that
$$\lambda(x_{i-1}\lessdot x_i)<\lambda(x_i\lessdot x_{i+1})<\lambda(x_{i+1}\lessdot x_{i+2}).$$

\begin{definition}\label{definition:braidrelation}
Let $P$ be a graded poset and let $\lambda$ be an ER-labeling of $P$ with the rank two switching
property. For every saturated chain $\c \in \mathcal{M}_{[x,y]}$ with a critical condition at rank $i$ we
can obtain saturated chains $\c^{\prime}$ and $\c^{\prime\prime}$ by removing the ascents at ranks
$i$ and $i+1$ by a sequence of exchanges $\c
\stackrel{U_i}{\rightharpoonup} \c_1 \stackrel{U_{i+1}}{\rightharpoonup} \c_2
\stackrel{U_i}{\rightharpoonup}
\c^{\prime}$ and $\c
\stackrel{U_{i+1}}{\rightharpoonup} \c_3 \stackrel{U{i}}{\rightharpoonup}\c_4
\stackrel{U_{i+1}}{\rightharpoonup}
\c^{\prime \prime}$.
We say that $\lambda$ satisfies the \emph{braid relation}  if for every such  $\c \in \mathcal{M}_{[x,y]}$
we have
that $\c^{\prime}=\c^{\prime\prime}$.
In
other words, we have $U_{i}U_{i+1}U_{i}(\c) = U_{i+1}U_{i}U_{i+1}(\c)$ for chains $\c$ which have a
critical condition at rank  $i$.
\end{definition}

Note that by the discussion above, if the ER-labeling $\lambda$ satisfies the braid relation then
the graph $G_{[x,y]}$ is locally confluent, hence we have  the following corollary.

\begin{corollary}\label{corollary:uniquesink}
 Let $P$ be a graded poset and let $\lambda$ be an ER-labeling of $P$ satisfying
 \begin{itemize}
  \item the rank two switching property, and
  \item the braid relation.
 \end{itemize}

 Then for every interval $[x,y]$ in $P$, we have that
each connected component of $G_{[x,y]}$ has a unique sink, i.e., a unique ascent-free saturated
chain.
\end{corollary}

\subsubsection{Quotient posets}
We now turn our attention to quotient posets, which is the other main tool we use for the construction of Whitney duals.  We begin with a definition.

\begin{definition}\label{definition:quotientposet}
Let $P$ be a graded poset and let $\sim$ be an equivalence relation on $P$ such that if $x\sim y$,
then
$\rho(x) = \rho(y)$.  We define the \emph{quotient poset} $P/\sim$ to be the set of equivalence
classes ordered by  $X\leq Y$ if and only if there exists $x\in X$, $y\in Y$ and $z_1,z_2, \dots,
z_k\in P$ such that
\begin{equation}\label{equation:equivalencerelation}
x= z_0\leq z_1\sim z_2 \leq\cdots\leq z_{n-1}\sim z_k\leq z_{k+1}= y.
\end{equation}
\end{definition}

For any element $x\in P$ we will denote by $[x]$ its corresponding equivalence
class in $P/\sim$.
The next proposition follows from Definition \ref{definition:quotientposet}.

\begin{proposition} \label{proposition:quotientposet}
Let $P$ be a graded poset and let $\sim$ be an equivalence relation on $P$ such that if $x\sim y$,
then
$\rho(x) = \rho(y)$.  Then we have the following.
\begin{enumerate}
\item $P/\sim$ is a poset.
\item For $X,Y\in P/\sim$,  $X\lessdot Y$ if and only if $x\lessdot y$ for some $x\in X$ and
$y\in Y$.
\item $P/\sim$ is graded and for $X\in P/\sim$, we have $\rho(X) = \rho(x)$ for all $x\in X$.

\end{enumerate}
\end{proposition}

\begin{proof}Part (2) can be easily verified from the definitions and part (3) is a consequence of
parts (1) and (2) since they imply that the function $\rho(X) = \rho(x)$, for
an arbitrary $x\in X$, is a well-defined rank function.
We show that $P/\sim$ together with the relation $\leq$ satisfies the three properties of a poset.
\begin{enumerate}
\item (Reflexive) This is clear.
\item (Antisymmetric) Suppose that $X\leq Y$ and $Y\leq X$.  Since elements in each
equivalence class have the same rank, then for any elements $x\in X$ and $y \in Y$
we have that $X\leq Y$ implies $\rho(x) \leq \rho(y)$.  Similarly, $Y\leq X$ implies
$\rho(y)\leq
\rho(x)$ and hence $\rho(x)=\rho(y)$.  Thus in (\ref{equation:equivalencerelation}) none of the
inequalities can
be strict, implying $x\sim y$ and hence $X=Y$.
\item (Transitive)  Suppose that $X\leq Y$ and $Y\leq Z$.  Then by the definition of $\le$ in
(\ref{equation:equivalencerelation}), $X\leq Y$ implies that there exist $x\in X$, $y\in Y$ and
$u_1,\dots, u_{k}$ such
that
\begin{align}\label{equation:chain1}
 x= u_0\leq u_1\sim u_2 \leq\cdots\leq u_{k-1}\sim u_k\leq u_{k+1}= y.
\end{align}
Also $Y\leq Z$ implies that there are $y'\in Y$, $z\in Z$ and $w_1,\dots, w_s$ such that
\begin{align}\label{equation:chain2}
 y'=w_0\leq w_1\sim w_2 \leq \cdots\leq w_{n-1}\sim w_n\leq w_{n+1}= z.
\end{align}
Since both $y$ and $y'$ are in $Y$, we have that $y\sim y'$. This together
with (\ref{equation:chain1}) and (\ref{equation:chain2}) imply that $X\leq
Z$.\qedhere
\end{enumerate}
\end{proof}

The poset $P/\sim$ with the relation defined above is called the \emph{quotient
poset} of $P$ by the relation $\sim$.

\begin{definition}\label{definition:lambdarelation}
Given a poset $P$, let $C(P)$ denote the poset whose elements are saturated chains
of $P$  starting at $\hat{0}$ ordered by inclusion. We call $C(P)$ the \emph{chain poset of
$P$}.
Figure~\ref{fig:chainPoset} depicts $\Pi_3$ and $C(\Pi_3)$.    If $\c\in C(P)$, we write $e(\c)$
for
the
maximal element of $\c$, i.e., the element of $P$ where $\c$ terminates. Suppose that $\lambda$ is
an
ER-labeling of $P$ with the rank two switching property.
Let $\sim_\lambda$ be the equivalence relation on $C(P)$ defined by $\c_1 \sim_\lambda \c_2$
whenever, $\c_1$ and $\c_2$ are in the same connected component
of
$G_{[\hat{0},e(\c_1)]}$.  We will use $Q_\lambda(P)$ to denote $ C(P)/\sim$.

\end{definition}

Note that by the nature of the quadratic exchanges, for every  $X \in Q_\lambda(P)$  and $\c,\c'\in
X$ we have that $e(\c)=e(\c')$.  Thus,  we can also define $e(X)=e(\c)$ for any
$\c \in X$.

\begin{figure}
\begin{subfigure}{0.25\textwidth}
                \centering
\input{pictures/partition_poset_n3_with_labels.tikz}

		  \caption{$\Pi_3$ with labeling $\lambda$}
                  \label{fig:examplePi32}

\end{subfigure}~\hspace{-10 pt}
\begin{subfigure}{0.5\textwidth}
                \centering
\input{pictures/chains_in_partition_poset_n3.tikz}
                \caption{$C(\Pi_3)$}
                  \label{fig:exampleCPI3}

\end{subfigure}~	\hspace{-80  pt}
\begin{subfigure}{.5\linewidth}
                \centering

\input{pictures/quotient_poset_partitions_n3.tikz}
 \caption{$Q_\lambda(\Pi_3)$ with labeling $\lambda^*$}
                  \label{fig:exampleQPi}

\end{subfigure}~

         \caption{}
  \label{fig:chainPoset}

\end{figure}

\begin{example} Consider again $\Pi_3$ and its chain poset $C(\Pi_3)$ shown in
Figure~\ref{fig:exampleCPI3}. Every element of $C(\Pi_3)$ is in its own equivalence class
except for the chains
$1/2/3\lessdot 12/3\lessdot 123$ and $1/2/3\lessdot 13/2\lessdot 123$ since
$1/2/3\lessdot 12/3\lessdot 123 \stackrel{U_1}{\rightharpoonup} 1/2/3\lessdot 13/2\lessdot 123$ is the only
directed edge in $G_{[1/2/3, 123]}$. Taking the quotient to obtain $Q_\lambda(\Pi_3)$, we get the
poset in Figure~\ref{fig:exampleQPi}, where we have  identified the equivalence classes by the
underlying set of labels on the chains.  By comparing with Figure~\ref{figure:exampleER}, one can
observe that $\mathcal{ISF}_3$ and $Q_\lambda(\Pi_3)$ are isomorphic.
\end{example}

Note that by the definition of $\sim_{\lambda}$, if $\lambda$ satisfies the conditions of Corollary
\ref{corollary:uniquesink} then each equivalence class $X \in
Q_{\lambda}(P)$ contains a unique ascent-free maximal chain in $[\hat{0},e(X)]$. In fact this is a
correspondence between ascent-free saturated chains starting at $\hat{0}$ in $P$ of
length $k$ and equivalence classes in $Q_\lambda(P)$ of rank $k$. Hence, using Proposition
\ref{proposition:ERwhitneynumbers} and the definition of
Whitney numbers of the second kind in Equation \ref{definition:whitneysecondkind} we are able
to conclude at this point that $|w_k(P)| = W_k(Q_\lambda(P))$.

We now turn our
attention to the task of satisfying the other half of Definition \ref{definition:whitneyduals}, that is, we
would like to have in addition that  $W_k(P) = |w_k(Q_\lambda(P))|$.  This will allow us
to conclude that
$Q_\lambda(P)$ is a Whitney Dual
of $P$. Our strategy will be to define an edge labeling $\lambda^*$ on $Q_\lambda(P)$ that under
certain conditions is an ER$^*$-labeling. We will then show that the saturated
chains from $\hat{0}$ in $Q_{\lambda}(P)$ under the newly defined ER$^*$-labeling and the ones
in $P$ under the labeling $\lambda$ are in a label-preserving bijection. This together with
Proposition \ref{proposition:ERwhitneynumbers} imply that $P$ and $Q_{\lambda}(P)$ are Whitney
duals.

To define this labeling recall that, by  definition,
$\c_1 \sim_\lambda\c_2$ implies  $S(\c_1)= S(\c_2)$.   In
light of this, we will use $S(X)$ to denote the multiset of labels in any chain in $X$. Moreover,
if $X\lessdot Y$ in  $Q_\lambda(P)$  then there exists a unique element in $S(Y)\setminus S(X)$.
Define an edge labeling $\lambda^*$ on $Q_\lambda(P)$ by
\begin{equation}\label{equation:lambdastar}
\lambda^*(X\lessdot Y)= S(Y)\setminus S(X).
\end{equation}
 This edge labeling for $Q_\lambda(\Pi_3)$ appears in
Figure~\ref{fig:exampleQPi}.  We will now consider a pair of structural lemmas that will be useful
in the
following discussion.

\begin{lemma}\label{lemma:main}
 Let $X_1\lessdot X_2 \lessdot \cdots \lessdot X_k$ be a saturated chain in $Q_\lambda(P)$.
\begin{enumerate}
\item [(a)] We have that $e(X_1)\lessdot e(X_2) \lessdot \cdots \lessdot e(X_k)$  is a saturated chain in
\item[(b)] For any chain $\c \in X_1$ we have that $\c \cup \{e(X_2),\dots,e(X_i)\} \in X_i$ for
all $i=1,\dots k$.
\item[(c)]  If $X_1=X_1'\lessdot X_2'\lessdot\cdots \lessdot X_k'$ is another saturated chain with
$e(X_i) = e(X_i')$ for all $i$, then $X_i = X_i'$ for all $i$.

\end{enumerate}
\end{lemma}
\begin{proof}
First we show (a) holds. Consider the cover $X_{i-1}\lessdot X_i$.  By
Proposition~\ref{proposition:quotientposet}, there exists a $\c \in X_{i-1}$ and $\d\in X_{i}$ with
$\c \lessdot \d$ in $C(P)$.   By the definition of the poset $C(P)$ we have that  $\d= \c\cup
\{e(X_i)\}$ and so $e(X_{i-1}) =
e(\c)\lessdot
e( \d) = e(X_i)$.  Thus, $e(X_1)\lessdot e(X_2)\lessdot \cdots \lessdot e(X_k)$ is a saturated chain in
$[e(X_1), e(X_k)]$.

To show (b), we use induction on $k$. The case when $k=1$ is trivial. Now suppose $k \geq 2$.  By
induction, for any  $\c \in X_1$ we have $\c \cup \{e(X_2),\dots,e(X_{i})\}
\in X_{i}$ for $i \le k-1$. We want to show that $\c \cup \{e(X_2),\dots, e(X_{k-1}),e(X_k)\} \in X_k$.  First,
note that by part (a), $\c \cup \{e(X_2),\dots,e(X_{k-1}), e(X_k))\}$ is a saturated chain in $P$. Now
since
$X_{k-1}\lessdot X_{k}$,  Proposition~\ref{proposition:quotientposet} implies that there exists $\d
\in X_{k-1} $ and $\d'\in X_k$ with $\d' = \d \cup\{e(X_k)\}$.  Since  $\c \cup
\{e(X_2),\dots,e(X_{k-1})\}$ and $\d$ both belong to $X_{k-1}$,  we know these two chains are
related by a sequence of
quadratic moves.  It follows that $\c \cup \{e(X_2),\dots,e(X_{k-1}), e(X_{k})\}$ and $\d'$ are
related by the exact same moves and so are in the same equivalence class.  Thus, $\c \cup
\{e(X_2),\dots,e(X_{k-1}), e(X_k))\}\in X_k$.

Note that part (c) follows directly from part (b).
\end{proof}

\begin{lemma}\label{lemma:saturatedchainssamelabelsinPandQ}
 Let $\mathbf{C}:(X_1\lessdot X_2 \lessdot \cdots \lessdot X_k)$ be a saturated chain in $Q_\lambda(P)$ and  let
$\c:(e(X_1)\lessdot e(X_2) \lessdot \cdots \lessdot e(X_k))$ be the corresponding saturated chain in $P$.
The
words of labels of these two saturated chains under their respective labelings are equal, i.e.,
$\lambda^*(\mathbf{C})=\lambda(\c)$.
\end{lemma}
\begin{proof}
 By Equation~\ree{equation:lambdastar}, $\lambda^*(X_{i-1}\lessdot X_{i})=
S(X_i)\setminus S(X_{i-1}) = S(\d')\setminus S(\d) = \lambda(e(X_{i-1})\lessdot e(X_i))$, where
$\d \in X_{i-1}$ and $\d' \in X_{i}$ are such that $\d \lessdot \d'$.
\end{proof}

To prove that the labeling $\lambda^*$ given in  Equation (\ref{equation:lambdastar}) is an
ER${}^*$-labeling  of $Q_\lambda(P)$ we will need the additional following condition.

\begin{definition}\label{definition:cancellative}
Let $\lambda$ be an ER-labeling on $P$ with the rank two switching property. We say
that $\lambda$ is \emph{cancellative} if for every $z<x<y$
in $P$,
$\c\in \mathcal{M}_{[z,x]}$ and $\c_1,\c_2 \in \mathcal{M}_{[x,y]}$ we have that
$$\c\cup \c_1 \sim_{\lambda} \c \cup \c_2 \text{ implies }  \c_1 \sim_{\lambda} \c_2.$$
\end{definition}

\begin{proposition}\label{proposition:QERStarLabel}
Let $P$ be a graded poset and let $\lambda$ be an ER-labeling of $P$ satisfying
 \begin{itemize}
  \item the rank two switching property,
  \item the braid relation, and
  \item the cancellative property.
 \end{itemize}

Then the labeling $\lambda^*$ of $Q_\lambda(P)$ given by  Equation~\ree{equation:lambdastar} is an
ER$^*$ labeling.
\end{proposition}
\begin{proof}
Let $[X,Y]$ be an interval in $Q_{\lambda}(P)$.  We will show that there is a unique
ascent-free maximal chain in $[X,Y]$.

{\bf The interval $[X,Y]$ contains an ascent-free maximal chain.}
Pick
any $\c \in X$ and let $X=X_1\lessdot X_2\lessdot \cdots \lessdot X_k=Y$ be a maximal chain in $[X,Y]$.
By
Lemma~\ref{lemma:main} parts (a) and (b), $e(X_1)\lessdot e(X_2) \lessdot \cdots \lessdot
e(X_k)$ is a saturated chain in $[e(X), e(Y)]$ and $\c\cup \{e(X_2), e(X_3),\dots, e(X_k)\} \in Y$.
The chain  $e(X_1)\lessdot
e(X_2) \lessdot \cdots \lessdot e(X_k)$
may not be ascent-free in $[e(X), e(Y)]$, but it is related to one by a sequence of quadratic
exchanges.    Suppose
the related ascent-free chain is $e(X)=x_1'\lessdot x_2'\lessdot \cdots \lessdot x_k'=e(Y)$.   For each $1\leq i \leq k$, let $X_i'$
be the equivalence class containing $\c \cup \{x_2',x_3', \dots, x_i'\}$.  Then it must be the
case that $X_k' = Y$ since we are using quadratic exchanges on $\c\cup \{e(X_2), e(X_3),\dots,
e(X_k)\}\in Y$.
In $C(P)$ we have that $\c \lessdot \c \cup\{x_2'\} \lessdot
\cdots\lessdot \c \cup \{x_2',x_3', \dots, x_k'\}$ and hence, by the quotient
poset definition of $Q_{\lambda}(P)$ we have that $X=X_1'\lessdot X_2'\lessdot \cdots
\lessdot X_k'=Y$ is a maximal chain in $[X,Y]$.
 Moreover, by  Lemma~\ref{lemma:saturatedchainssamelabelsinPandQ}, the labels along this
chain are the same as along
$x_1'\lessdot x_2'\cdots \lessdot x_k'$.  It follows that $[X,Y]$ has an ascent-free maximal chain.

{\bf The ascent-free maximal chain found above is unique.}
Suppose this
was not the case and that $X=X_1\lessdot X_2\lessdot \cdots \lessdot X_k=Y$ and $X=X_1'\lessdot X_2'\lessdot
\cdots \lessdot X_k'=Y$ are both ascent-free maximal chains in $[X,Y]$.  Pick any $\c \in X$.  Then
by  Lemma~\ref{lemma:main} part (b), $\c\cup \{e(X_2), e(X_3),\dots, e(X_k)\}$ and $\c\cup
\{e(X_2'), e(X_3'),\dots, e(X_k')\}$ are both chains in $Y$. By the cancellative property $\c\cup
\{e(X_2), e(X_3),\dots, e(X_k)\} \sim \c\cup \{e(X_2'), e(X_3'),\dots, e(X_k')\}$ implies that
$e(X_1)\lessdot e(X_2)\lessdot \cdots \lessdot e(X_k)$ and $e(X_1')\lessdot e(X_2') \lessdot \cdots \lessdot e(X_k')$
are related by quadratic exchanges.  It follows that both of these chains are in the same
connected component of $G_{[e(X),e(Y)]}$ and by Lemma~\ref{lemma:saturatedchainssamelabelsinPandQ},
they are both ascent-free.  However, Corollary~\ref{corollary:uniquesink}, asserts  that there is a
unique
ascent-free maximal chain  in each connected component implying that $e(X_i) = e(X_i')$ for
all $i$.   Applying Lemma~\ref{lemma:main} part (c) we conclude that $X_i =X_i'$ for all $i$.  It
follows that there is a unique ascent-free maximal chain in each interval  and so $\lambda^*$ is an
ER$^*$ labeling.
\end{proof}

Before stating the main theorem of this section let us discuss a very important class of labelings
that satisfy the conditions of Proposition \ref{proposition:QERStarLabel}.

\begin{theorem}\label{theorem:standardEWlabelings}
Let $\lambda$ be an $ER$-labeling satisfying
 \begin{itemize}
  \item the rank two switching property, and
  \item in each interval each ascent-free maximal chain has a unique word of labels;
 \end{itemize}
 then $\lambda$ satisfies
  \begin{itemize}
  \item the rank two switching property,
  \item the braid relation, and
  \item the cancellative property.
 \end{itemize}
\end{theorem}
\begin{proof}
 We need to check that $\lambda$ satisfies the braid relation and the cancellative property.

{\bf The braid relation is satisfied by $\lambda$.}
Indeed, let $\c$ be a saturated chain that has a
critical condition at rank $i$.  We obtain saturated chains $\c^{\prime}$ and $\c^{\prime\prime}$
by
removing the ascents at ranks $i$ and $i+1$ by a sequence of exchanges $\c
\stackrel{U_i}{\rightharpoonup} \c_1 \stackrel{U_{i+1}}{\rightharpoonup} \c_2
\stackrel{U_i}{\rightharpoonup}
\c^{\prime}$ and $\c \stackrel{U_{i+1}}{\rightharpoonup} \c_3 \stackrel{U_i}{\rightharpoonup} \c_4
\stackrel{U_{i+1}}{\rightharpoonup}
\c^{\prime \prime}$. By the definition of a quadratic exchange we have that
$\c^{\prime}$ and $\c^{\prime\prime}$, when restricted to the interval between  ranks $i-1$ and
$i+2$, have the same ascent-free word of labels and
hence $\c^{\prime}=\c^{\prime\prime}$.

{\bf The cancellative property is satisfied by $\lambda$.} We are going to prove, using induction on the value of
$\rho(x)-\rho(z)$, that for every $z<x<y$ in $P$, $\c\in \mathcal{M}_{[z,x]}$ and $\d,\d' \in \mathcal{M}_{[x,y]}$ we
have that $\c\cup \d \sim_{\lambda} \c \cup \d' \text{ implies }  \d \sim_{\lambda} \d'.$

When $\rho(x)-\rho(z)=1$ we have  $\c:(z\lessdot x)$ and without loss of generality we can
assume,
perhaps after applying enough quadratic exchanges, that $\d$ and $\d'$ are ascent-free. Hence
the only possible ascents must happen in the step $z\lessdot x \lessdot d_1$ for $\c \cup \d$ and in
the
step $z\lessdot
x \lessdot d'_1$ for $\c\cup \d'$. Note that then quadratic exchanges on $\c\cup \d$ can only shuffle
the label $\lambda(z \lessdot x)$ across the word $\lambda(\d)$ and quadratic exchanges on
$\c\cup \d'$
can only shuffle $\lambda(z \lessdot x)$ across the word $\lambda(\d')$. Because we have $\c\cup \d
\sim_{\lambda} \c
\cup \d'$ and the braid relation, Corollary \ref{corollary:uniquesink} implies that the ascent-free word of labels obtained after all the quadratic exchanges have been
applied to both $\c\cup \d$ and $\c\cup \d'$ is the same. Hence we originally had $\lambda(\d)=\lambda(\d')$. Uniqueness
of the ascent-free word of labels implies then $\d=\d'$.

Now consider the case when $\rho(x)-\rho(z)=k>1$ and we have chains $\c\in \mathcal{M}_{[z,x]}$ and
$\d,\d' \in \mathcal{M}_{[x,y]}$ such that $\c\cup \d \sim_{\lambda} \c \cup \d'$. Note first that
$z<c_{k-1}<y$, and we will consider instead the saturated chains $\c^{\prime}:=\c\setminus \{x\}
\in \mathcal{M}_{[z,c_{k-1}]}$, $\mathbf{m}:=\{c_{k-1}\}\cup
\d \in \mathcal{M}_{[c_{k-1},y]}$ and $\mathbf{m}^{\prime}=\{c_{k-1}\}\cup \d' \in \mathcal{M}_{[c_{k-1},y]}$.  We have that
$\rho(c_{k-1})-\rho(z)=\rho(x)-\rho(z)-1$ and $\c^{\prime}\cup \mathbf{m}=\c\cup \d \sim_{\lambda} \c
\cup \d'=\c^{\prime}\cup \mathbf{m}^{\prime}$, hence by induction we conclude that $\mathbf{m}
\sim_{\lambda} \mathbf{m}^{\prime}$. Now if
we consider $c_{k-1}\lessdot x < y$, the argument above says that$\{c_{k-1}\lessdot x\}\cup \d=
\mathbf{m} \sim_{\lambda} \mathbf{m}^{\prime}=\{c_{k-1}\lessdot x\}\cup \d'$ in $\mathcal{M}_{[c_{k-1},y]}$ and since
$\rho(x)-\rho(c_{k-1})=1$ we are back in the base case that implies $\d \sim_{\lambda} \d'$.
\end{proof}

\begin{remark}
Note that in  Theorem \ref{theorem:standardEWlabelings} we can replace the second condition by the stronger requirement that  maximal chains have unique word of labels.
\end{remark}

We are now in a position to provide names to the
type of edge labelings that allow us to construct Whitney duals. We call these labelings
EW-labelings, where the letter ``W" comes from  the fact that they provide sufficient conditions to
construct Whitney duals.

\begin{definition}\label{definition:EW}
Let $\lambda:\mathcal{E}(P)\rightarrow \Lambda$ be an ER-labeling of $P$.  We
say $\lambda$ is an \emph{EW-labeling} if it satisfies
 \begin{itemize}
  \item the rank two switching property, and
  \item in each interval each maximal chain has a unique word of labels.
 \end{itemize}
\end{definition}

\begin{definition}\label{definition:generalizedEW}
Let $\lambda:\mathcal{E}(P)\rightarrow \Lambda$ be an ER-labeling of $P$.  We
say $\lambda$ is a \emph{generalized EW-labeling} if it satisfies
 \begin{itemize}
  \item the rank two switching property,
  \item the braid relation, and
  \item the cancellative property.
 \end{itemize}
\end{definition}

The following proposition provides an insight into the construction process that generates
$Q_{\lambda}(P)$ from $P$ using $\lambda$. According to Proposition
\ref{proposition:bijectionbetweensaturatedchains}, we can   think  of $Q_{\lambda}(P)$
as a poset that is obtained from $P$ by ``pulling apart" saturated chains
from $\hat{0}$.

\begin{proposition} \label{proposition:bijectionbetweensaturatedchains}
Let $\lambda$ be a generalized EW-labeling
of $P$.  There is a label preserving bijection from the set of saturated
chains from $[\hat{0}]$ of length $k$ in $Q_{\lambda}(P)$ and the set of saturated chains from
$\hat{0}$ of length $k$ in $P$. In particular, there is a label preserving bijection
$\mathcal{M}_{Q_{\lambda}(P)}\rightarrow \mathcal{M}_{P}$ between maximal
chains.
\end{proposition}

\begin{proof}
Fix $k$ and let $S_{Q,k}$ be the set of saturated
chains from $[\hat{0}]$ of length $k$ in $Q_{\lambda}(P)$ and $S_{P,k}$ be the set of saturated
chains
from $\hat{0}$ of length $k$ in $P$. Let
$\varphi : S_{Q,k} \rightarrow  S_{P,k}$  be defined by $$\varphi([\hat{0}]=X_0\lessdot
X_1\lessdot \cdots \lessdot X_k) = (\hat{0}= e(X_0)\lessdot e(X_1)\lessdot \cdots \lessdot e(X_k)).$$
By Lemma~\ref{lemma:main} part (a) we know $\varphi$ is well-defined and by
Lemma~\ref{lemma:saturatedchainssamelabelsinPandQ} we know that $\varphi$ preserves the word of
labels.  By
Lemma~\ref{lemma:main} part (c),  $\varphi$ is injective.

Finally, we show that $\varphi$ is  surjective.    Let $\hat{0}=x_0\lessdot x_1\lessdot \cdots \lessdot
x_k \in S_{P,k}$.  Let $\c_i = \{x_0,x_1,\dots, x_i\}$, then by definition $\c_0\lessdot
\c_1\lessdot \cdots \lessdot\c_k$ is in $C(P)$.  By the definition of a quotient poset,
$[\c_0]\lessdot [\c_1]\lessdot
\cdots \lessdot  [\c_k]$ is in $S_{Q,k}$ and it  is a preimage of
$\hat{0}=x_0\lessdot
x_1\lessdot \cdots \lessdot x_k$.
\end{proof}

Definition  \ref{definition:whitneyduals}
together with Propositions \ref{proposition:ERwhitneynumbers},
\ref{proposition:QERStarLabel}  and \ref{proposition:bijectionbetweensaturatedchains}
imply our main theorem, that we are ready to state in the
language of Definitions \ref{definition:EW} and \ref{definition:generalizedEW}.

\begin{theorem}\label{theorem:EW}
Let $P$ be a graded poset with a generalized EW-labeling $\lambda$.  Then
$Q_\lambda(P)$ is a Whitney dual of $P$.
\end{theorem}

\begin{remark}In \cite{DleonHallam2017} the authors defined the related concept of
an $\overline{\mbox{EW}}$-labeling. The reason for the use of an overline in that article is that
the conditions on
those labelings are more restrictive but imply the conditions of
Definition
\ref{definition:EW}.
While the definition of
an  $\overline{\mbox{EW}}$-labeling greatly simplifies the proofs of the theorems we have presented
here, there are posets with EW-labelings, but no known  $\overline{\mbox{EW}}$-labeling.   See
Section~\ref{weightedPartSec} for some examples.

\end{remark}

\subsection{Chain-edge labelings}\label{section:CWlabelings}

We show in this subsection that the definitions and constructions given for EW-labelings also extend to the
generality of chain-edge labelings with the same consequences with respect to Whitney duality.

\begin{definition}
Let $\mathcal{M}\mathcal{E}(P)$ denote the set of pairs $(\mathbf{m},e)$ where $\mathbf{m}$ is a maximal chain in $P$ and $e$ is an
edge in the Hasse diagram of $\mathbf{m}$. A \emph{chain-edge labeling} or
\emph{C-labeling} of $P$ is a map $\lambda: \mathcal{M}\mathcal{E}(P)\rightarrow \Lambda$, where $\Lambda$ is some
poset of labels, satisfying the condition
that whenever two maximal chains coincide along the bottom $d$ edges then their labels also
coincide on these $d$ edges.
\end{definition}
\begin{definition}\label{definition:rooted_interval} A \emph{rooted interval} $[x,y]_{\r}$ in $P$
is a pair $([x,y],\r)$ where
$[x,y]$ is
an interval in $P$ and $\r$ is a saturated chain from
$\hat{0}$ to $x$.
\end{definition}

The rationale behind Definition \ref{definition:rooted_interval} is that in a C-labeling, when we
want to restrict to a smaller
interval $[x,y]$ in $P$, the labels depend on the initial saturated chain $\r$ from
$\hat{0}$ to $x$.

\begin{definition}
A C-labeling $\lambda$ of $P$ naturally induces a C-labeling $\lambda_{\r}$ in a rooted
interval $[x,y]_{\r}$ by letting the labels of a maximal chain $\c$ of $[x,y]$ be the ones
corresponding to the maximal chain $\r \cup \c$ in $[\hat{0},y]$. A  C-labeling is a
\emph{CR-labeling} if in every rooted interval $[x,y]_{\r}$ there is a unique increasing maximal
chain.
\end{definition}
It was shown by Bj\"orner and Wachs \cite{BjornerWachs1996} that a CR-labeling on a poset $P$
has the same implications with respect to M\"obius numbers as described in Theorem \ref{theorem:muER} in the case of an ER-labeling.
Hence we can describe the
Whitney numbers of a poset with a CR-labeling by the enumeration of
saturated chains in  the same way than Proposition \ref{proposition:ERwhitneynumbers}.
We have that $|w_k(P)|$ is the number of
ascent-free saturated chains starting at $\hat{0}$  of length $k$ and
$|W_k(P)|$ is the number of increasing saturated chains starting at $\hat{0}$ of length $k$ as
before.

\begin{definition}\label{definition:ranktwoC}
We say a C-labeling has the \emph{rank two switching property} if for every maximal chain
of the
form
$$\mathbf{m}:(\hat{0}= m_0 \lessdot m_1 \lessdot \cdots \lessdot m_{k})$$ with an
ascending step $\lambda(\mathbf{m}, m_{i-1}\lessdot m_i)< \lambda(\mathbf{m}, m_{i} \lessdot m_{i+1})$   at some
rank $i<k$ there is a unique
element $m'_i \neq m_{i}$  such that the chains $\mathbf{m}$ and
$$\mathbf{m}':(\hat{0}= m_0 \lessdot m_1  \lessdot \cdots \lessdot m_{i-1}\lessdot m'_i \lessdot m_{i+1}
\cdots\lessdot m_{k})$$
have the same word of labels except at rank $i$ where $\lambda(\mathbf{m}',m_{i-1}\lessdot m'_i)=
\lambda(\mathbf{m},m_{i}\lessdot m_{i+1})$ and
$\lambda(\mathbf{m}',m'_{i}\lessdot m_{i+1})=\lambda(\mathbf{m}, m_{i-1}\lessdot m_i)$. Moreover, the rank two switching property requires a consistency condition, that for any other
maximal
chain $\widetilde \mathbf{m}$ that coincides with $\mathbf{m}$ in the first $i+2$ elements ($\widetilde m_j=m_j$
whenever $j\le i+1$) the choice of the unique element also coincides, i.e., $\widetilde
m'_i=m'_i$.
\end{definition}

\begin{remark}
Note that in Definition \ref{definition:ranktwoC} there is an additional
consistency condition that is not present in Definition \ref{definition:ranktwoE}. This condition
guarantees that the restriction of the rank two switching property for intervals of the form
$[\hat{0},y]$ is well-defined when $y$ any  element of $P$ which is not necessarily maximal.
\end{remark}

In the situation of Definition \ref{definition:ranktwoC} we say that the chain
$\mathbf{m}'$, is obtained from $\mathbf{m}$ by a \emph{quadratic exchange at rank
$i$} and will  use the notation
$\mathbf{m}'=U_i(\mathbf{m})$. If $\mathbf{m}$ does not have an ascent at rank $i$ we define $U_i(\mathbf{m})=\mathbf{m}$.

As in the discussion after Definition \ref{definition:quadraticexchanges}, we define
graphs $G_{[x,y]_{\r}}$ given by quadratic exchanges but this time the elements are maximal chains
in
a rooted interval $[x,y]_{\r}$. To check confluency in $G_{[x,y]_{\r}}$ we invite the reader to
verify that by the consistency condition of the rank two switching property for C-labelings in
Definition \ref{definition:ranktwoC}, if a
maximal chain $\c$ has ascents at ranks $i$ and $j$ with $|i-j|>1$ it is necessarily true that
$U_iU_j(\c) = U_jU_i(\c)$.

We say that $\c : (\hat{0} =
x_0 \lessdot x_1 \lessdot \cdots \lessdot x_{\ell-1} \lessdot x_{\ell}= y)$ has a
\emph{double-ascent} or a \emph{critical condition at rank $i$} if
$$\lambda(\c, x_{i-1}\lessdot x_i)<\lambda(\c, x_i\lessdot x_{i+1})<\lambda(\c, x_{i+1}\lessdot
x_{i+2}).$$
For every saturated chain $\c$ from $\hat{0}$ with a critical condition at rank $i$ we say that
$\lambda$ satisfies the \emph{braid relation}  if we have $U_{i}U_{i+1}U_{i}(\c) =
U_{i+1}U_{i}U_{i+1}(\c)$.

When we have a  CR-labeling $\lambda$ satisfying the rank two switching property and the braid
relation, we ensure local confluency in $G_{[x,y]_{\r}}$ and hence, by Lemma \ref{lemma:newman}, the conclusion of Corollary
\ref{corollary:uniquesink} holds.  Therefore  each connected component of $G_{[x,y]_{\r}}$ has a
unique
sink, i.e., a unique ascent-free saturated
chain. We use the exact same definitions of chain
poset $C(P)$ and quotient poset
$Q_{\lambda}(P)$ given in Definition~\ref{definition:lambdarelation}; and also give the same
definition of the edge labeling $\lambda^*$ on $Q_\lambda(P)$ of Equation \ref{equation:lambdastar},
that is,
$$\lambda^*(X\lessdot Y)= S(Y)\setminus S(X). $$
Note that $\lambda^*$ is actually an E-labeling on
$Q_{\lambda}(P)$,
i.e. does not depend on maximal chains, even though the labeling $\lambda$ of $P$ is a C-labeling.
We
would like to conclude that $\lambda^*$ is also an ER$^*$-labeling in this case. To do this, we show
that the lemmas and propositions for
E-labelings that appeared in the previous subsection still hold in the C-labeling scenario.

It is straightforward to verify that Lemma \ref{lemma:main} is still valid in our new
setting, but we need a C-labeling analogue (Lemma
\ref{lemma:saturatedchainssamelabelsinPandQforClabeling} below) of Lemma
\ref{lemma:saturatedchainssamelabelsinPandQ} to be able to produce  an analogue of Proposition
\ref{proposition:bijectionbetweensaturatedchains}.

\begin{lemma}\label{lemma:saturatedchainssamelabelsinPandQforClabeling}
Let $\mathbf{D}:(X_1\lessdot X_2 \lessdot \cdots \lessdot X_k)$ be a saturated chain in $Q_\lambda(P)$ and  let
$\d:(e(X_1)\lessdot e(X_2) \lessdot \cdots \lessdot e(X_k))$  be the corresponding saturated chain
in $P$. Let $\c \in X_1$ and let $\lambda_{\c}$  be the induced labeling coming from $\lambda$ on
the rooted
interval $[e(X_1),e(X_k)]_{\c}$. Then the  words of labels of $\mathbf{D}$ and $\d$ under
their respective labelings are equal, i.e., $\lambda^*(\mathbf{D})=\lambda_{\c}(\d)$.
\end{lemma}
\begin{proof}
By Lemma \ref{lemma:main} we have that for all $i \le k$, $\c \cup
\{e(X_1),e(X_2),\cdots ,e(X_{i})\}\in X_{i}$. Then
by Equation~\ree{equation:lambdastar}, we have that
\begin{align*}
 \lambda^*(X_{i-1}\lessdot X_{i})&=
S(X_i)\setminus S(X_{i-1})\\
&= S(\c \cup \{e(X_1),e(X_2),\cdots ,e(X_{i})\})\setminus
S(\c\cup \{e(X_1),e(X_2),\cdots e(X_{i-1})\}) \\
&= \lambda(\c \cup \d,
e(X_{i-1})\lessdot e(X_i)).\qedhere
\end{align*}
\end{proof}

 \begin{definition}\label{definition:cancellativeC}
Let $\lambda$ be a CR-labeling on $P$ with the rank two switching property. We say
that $\lambda$ is \emph{cancellative} if for every $z<x<y$
in $P$, $\r \in \mathcal{M}_{[\hat{0},z]}$, $\c\in \mathcal{M}_{[z,x]}$ and $\c_1,\c_2 \in \mathcal{M}_{[x,y]}$ we have that
$$\c\cup \c_1 \sim_{\lambda_{\r}} \c \cup \c_2 \text{ implies }  \c_1 \sim_{\lambda_{\r \cup \c}}
\c_2.$$
\end{definition}

The reader can verify that the proof of Proposition \ref{proposition:QERStarLabel} is still valid
when $\lambda$ is a CR-labeling with the corresponding properties, replacing whenever necessary
intervals in $P$ by rooted intervals to take into account the labeling. We then have that a
CR-labeling $\lambda$ of $P$ with the rank two switching property, the braid relation and the
cancellative property induces the ER$^*$ labeling $\lambda^*$ of $Q_\lambda(P)$ given by
Equation~\ree{equation:lambdastar}.
\begin{definition}\label{definition:generalizedCWandCW}
A \emph{generalized CW-labeling}  $\lambda:\mathcal{M}\mathcal{E}(P)\rightarrow \Lambda$ is a CR-labeling  that
satisfies
 \begin{itemize}
  \item the rank two switching property,
  \item the braid relation, and
  \item the cancellative property.
 \end{itemize}

We say $\lambda$ is a \emph{CW-labeling} if it satisfies
 \begin{itemize}
  \item the rank two switching property, and
  \item in each rooted interval each ascent-free maximal chain has a unique word of labels.
 \end{itemize}
\end{definition}

 As with EW-labelings, it turns out in this scenario  that every CW-labeling is a generalized CW-labeling but we
do not know if the converse is true.

We are now ready to state the main theorem for (generalized)
CW-labelings that follows from Definition  \ref{definition:whitneyduals}
together with the C-labeling analogues of Propositions \ref{proposition:ERwhitneynumbers},
\ref{proposition:bijectionbetweensaturatedchains}, and
\ref{proposition:QERStarLabel}.

\begin{theorem}\label{theorem:CW}
Let $P$ be a graded poset with a (generalized) CW-labeling $\lambda$.  Then
$Q_\lambda(P)$ is a Whitney dual of $P$.
\end{theorem}

\section{The Whitney dual $Q_\lambda(P)$}\label{section:whitneydual}
In this section we first give a formula for the M\"obius function of
$Q_\lambda(P)$ given that $\lambda$ is a generalized CW-labeling. In the second part of the section
we provide a somewhat simpler description
of $Q_\lambda(P)$ given that $\lambda$ is a CW-labeling. This characterization only applies to
CW-labelings in the strict sense of the definition, so our description does not include all
generalized CW-labelings.

\subsection{The M\"obius function of $Q_{\lambda}(P)$}

We can characterize the M\"obius numbers of $Q_{\lambda}(P)$ using the fact that $\lambda^*$ of Equation \ref{equation:lambdastar} is
an ER${}^*$-labeling. An interesting fact is that $Q_{\lambda}(P)$ belongs to the famous family of posets whose M\"obius
numbers are $0$ or $\pm 1$. Hersh and M\'esz\'aros in \cite{HershMeszaros2017} have defined a
family
of edge labelings, that they coined SB-labelings, and that allowed them to conclude that a lattice
with
such a labeling has M\"obius numbers  $0$ or $\pm 1$. Their result partially answers a question
posed by  Bj\"orner  and Greene on why posets with these M\"obius values are plentiful in
combinatorics. It is still an open problem to find a characterization of when the posets
$Q_{\lambda}(P)$ are lattices. The family of posets $Q_{\lambda}(P)$  provide a
plethora of examples of posets whose M\"obius
numbers are $0$ or $\pm 1$.

Fix $x\in P$ and let $X^1, X^2,\dots, X^n$ be the different elements of $Q_\lambda(P)$ such that for all $i$, $e(X^i)=x$. Since there is exactly one increasing maximal chain
in $[\hat{0}, x]$, Definition \ref{definition:lambdarelation} implies that there is exactly one $X^i$
that contains this maximal increasing chain and all other $X^j$ for $j\neq i$ do not contain any
increasing maximal chain.   As we see in the next proposition it is exactly this class which has a
nonzero M\"obius value $\mu([[\hat{0}],X])$ in $Q_\lambda(P)$.

\begin{proposition}\label{proposition:mobiusQlambda}
 Let $\lambda$ be a generalized CW-labeling of $P$; $X,Y \in Q_{\lambda}(P)$ and
$\c\in X$.
 Then in $Q_\lambda(P)$ we have
$$
\mu([X,Y])= \begin{cases}
(-1)^{\rho(Y)-\rho(X)} & \mbox{if $Y$ contains $\c \cup \d$, where $\d$ is the unique
increasing }\\ & \mbox{maximal chain in $\mathcal{M}_{[e(X),e(Y)]_{\c}}$,}\\
0 & \mbox{otherwise.}
\end{cases}
$$
\end{proposition}
\begin{proof}
Since $\lambda$ is a CR-labeling, there exists a unique maximal chain $\d$ in
$[e(X),e(Y)]_{\c}$ which is increasing.  If $Y\in Q_\lambda(P)$ is the class that contains
$\c\cup\d$ then, by the definition of $Q_{\lambda}(P)$ and
Equation~\ree{equation:lambdastar},
there is an increasing saturated chain which terminates at $Y$ say $X=X_1\lessdot X_2\lessdot
\cdots \lessdot X_k=Y$.  Now suppose that there was another increasing chain $X=X_1'\lessdot
X_2'\lessdot \cdots \lessdot X_k'$ in $Q_\lambda(P)$ with $e(X_k')=e(Y)$.  Then by
Lemma~\ref{lemma:saturatedchainssamelabelsinPandQforClabeling} there would be a corresponding
maximal chain in
$[e(X), e(Y)]_{\c}$ which is increasing.  Since
there is a unique increasing maximal chain in $[e(X),e(Y)]_{\c}$, we know that $e(X_i) =e(X_i')$
for all $i$.  But (the CW-labeling version of) Lemma~\ref{lemma:main} part (c) implies that
$X_i=X_i'$ which is impossible.  Since (the CW-labeling version of)
Proposition \ref{proposition:QERStarLabel} asserts that $\lambda^*$
given in Equation (\ref{equation:lambdastar}) is an ER$^*$-labeling of $Q_{\lambda}(P)$ Theorem \ref{theorem:muER} gives the desired result.
\end{proof}
\begin{remark}
 Note that in the EW-version of Proposition \ref{proposition:mobiusQlambda} the chain $\c \in X$ is
irrelevant.
\end{remark}

\subsection{Another description of $Q_{\lambda}(P)$}

Let  $\Lambda$ be a poset and let $w$ be a word with letters in the alphabet $\Lambda$. Assume
that whenever $w_i<w_{i+1}$ we
are allowed to do exchanges on $w$ of the form
$$w_1w_2\cdots w_{i-1}w_iw_{i+1}w_{i+2}\cdots w_n \stackrel{i}{\rightarrow} w_1w_2\cdots
w_{i-1}w_{i+1}w_{i}w_{i+2}\cdots w_n.$$
It is not hard to check that this type of exchange produces a locally confluent relation and
after using Newman's Lemma \ref{lemma:newman} we can conclude that there is a unique ascent-free
word $w'$ that is related to $w$ in this manner. We define $\sort(w):=w'$ to be this unique
ascent-free word.  For example, if $\Lambda= \mathbb{Z}$, $w=85324$ then $\sort(w) = 85432$.

\begin{definition}\label{rLambdaDef}
Let $P$ be a poset with a CW-labeling $\lambda$. Let $R_\lambda(P)$ be the poset whose elements are
pairs $(x,w)$ where $x\in P$ and $w$ is the word of labels of an ascent-free saturated chain
$\c \in \mathcal{M}_{[\hat{0},x]}$ (note that by the definition of CW-labeling $w$ uniquely determines $\c$);
and such that $(x,w) \lessdot (y,u)$ whenever $x\lessdot y$ and $u = \sort(w \lambda(\c,x\lessdot
y))$ ($uv$ here means the concatenation of the words $u$ and $v$).
\end{definition}

\begin{example}

If we consider the EW-labeling of $\Pi_3$ given in
Example~\ref{example:labelingposetofpartitions}, we obtain the poset
$R_\lambda(\Pi_3)$ depicted in Figure~\ref{rLambdaPi3Fig}. Comparing $R_\lambda(\Pi_3)$  and
$Q_\lambda(\Pi_3)$ given in Figure~\ref{fig:exampleQPi}, we can observe directly
that these two posets are isomorphic.
\end{example}

\begin{figure}
\input{pictures/example_R_lambda.tikz}
\caption{$R_\lambda(\Pi_3)$} \label{rLambdaPi3Fig}
\end{figure}

\begin{theorem}\label{theorem:secondcaracterizationQlambda}
If $\lambda$ is a CW-labeling of $P$, then $R_\lambda(P)\cong Q_\lambda(P)$.
\end{theorem}
\begin{proof}
Let $\varphi: R_\lambda(P) \rightarrow Q_\lambda(P)$ be given by $\varphi((x,w)) = [\c]$
where $\c \in \mathcal{M}_{[\hat{0},x]}$ is the unique ascent-free chain determined by $w$. Using the fact
that when $\lambda$ is a CW-labeling each equivalence class in $Q_\lambda(P)$ contains a unique
ascent-free saturated chain, one can see that $\varphi$ is a well-defined bijection. We want to see
that $\varphi$ and $\varphi^{-1}$ are poset maps.

We now show that $\varphi$ is an order-preserving map. Since we are working with finite posets it is enough to
show that   $\varphi$ preserves cover relations. Suppose that $(x,w) \lessdot (y,u)$ in
$R_\lambda(P)$. Let $\c$ and $\d$ be  the unique saturated chains from $\hat{0}$
determined by $w$ and $u$, respectively. Then  $\varphi((x,w)) = [\c]$ and $\varphi((y,u)) = [\d]$.
Let $\d'=\c \cup \{y\}$, Since $\d'$ is a saturated chain from $\hat{0}$ to $y$ with word of
labels $w \lambda(\c\cup \{y\},x\lessdot y)$, we can use the rank two switching property to see that $\d'$
is
equivalent to an ascent-free chain with labels $\sort(w \lambda(\c, x\lessdot y))=u$. Since $u$
determines a unique saturated chain we have that  $\d' \sim \d$.  Moreover, $\c \lessdot \d'$ in
$C(P)$, so we have $\varphi((x,w))= [\c] \lessdot [\d] =\varphi((y,u))$.

Now we show $\varphi^{-1}$ is also an order-preserving map. To see why, suppose that $X \lessdot
Y$ in $Q_{\lambda}(P)$; and let $x=e(X)$, $y= e(Y)$, $\varphi^{-1}(X)=(x,w)$ and
$\varphi^{-1}(Y)=(y,u)$.  By the definition of
$Q_\lambda(P)$ and Proposition \ref{proposition:quotientposet}, there are chains $\c \in X$ and
$\d'\in Y$ such that $\d'=\c \cup \{y\}$ and hence $x\lessdot y$. We may assume, without loss of generality,
that $\c$ is the ascent-free chain in $X$ with word $w$ (otherwise, apply quadratic exchanges until
you obtain an ascent-free chain in $X$).
Note that $w\lambda(\d',x\lessdot y)$ is the word of
labels of $\d'$. Let $\d$ be the ascent-free chain in $Y$ with word of labels $u$. Since $\d \sim
\d'$ we also have that the word of labels of $\d$ is $\sort(w\lambda(\c,x\lessdot y))$ and so
$u=\sort(w\lambda(\c,x\lessdot y))$. We obtain then that $\varphi^{-1}(X)=(x,w)\lessdot (y,u)=
\varphi^{-1}(Y)$ as desired.
\end{proof}

The new characterization of $Q_\lambda(P)$ that was given  in Theorem
\ref{theorem:secondcaracterizationQlambda} can be helpful providing combinatorial descriptions of
Whitney duals (see Section \ref{section:noncrossingpartitiondual}).

\section{Examples of posets with Whitney labelings}\label{section:examplesWposets}
In this section we give several examples of posets with Whitney labelings.  By
Theorem~\ref{theorem:CW},
this implies that these posets also have Whitney duals.
\subsection{Geometric lattices}

In \cite{Stanley1974} Stanley introduced an edge labeling for geometric lattices that is an
ER-labeling (In fact, as shown by Bj\"orner in \cite{Bjorner1982}  it is an EL-labeling).   We give
the
definition
below and then show it is also an EW-labeling.

\begin{definition}\label{geomLabDef}
Let $L$ be a geometric lattice with set of atoms $A(L)$.  Fix a total order on $A(L)$.  Now define
$\lambda:\mathcal{E}(L) \rightarrow A(L)$ by  setting $\lambda(x\lessdot y)=a$ where $a$ is the smallest atom
such that $x\vee a = y$.  We call $\lambda$ a \emph{minimum labeling of $L$}. Note that this
labeling can be different for different total orders on $A(L)$
\end{definition}

\begin{example}\label{example:labelingPIn}
The labeling $\lambda$ of $\Pi_n$ in Example~\ref{example:labelingposetofpartitions} is  a minimum
labeling.
Here we associate each atom of $\Pi_n$ with the ordered pair $(i,j)$ where $ij$ is the unique
nontrivial block in the atom and such that $i<j$.  We then order the atoms lexicographically.
\end{example}

\begin{proposition}\label{rk2Geom}
For any geometric lattice $L$ a minimum labeling of $L$ is an
EW-labeling.
\end{proposition}
\begin{proof}
It was shown in~\cite{Stanley1974} that a minimum labeling is an ER-labeling.  Also, for any
interval $[x,y]$
the labels along any maximal chain uniquely determine the chain since one can read off the elements
of the chain by taking joins of $x$ with the labels along the chain.  Thus it suffices to
show that a minimum labeling has the rank two switching property.

Let
$\lambda$ be a minimum labeling of $L$, let $[x,y]$ be an interval of rank two and suppose that
$ij$ is the word of labels of the increasing chain,  $x\lessdot x\vee i \lessdot x\vee i\vee j =y$.
Since $L$ is geometric and $j$ is an atom not underneath $x$, $x\lessdot x\vee j\lessdot y$.  Observe
that
 $\lambda(x\lessdot x\vee j) =j$, since if this was  not the case, this would imply $\lambda(x\vee i
\lessdot y) <j$ which is a contradiction.
  Moreover,  $i$ is not below $x\vee j$  and $i$  is below $y$.   Since there is a unique
increasing
chain in $[x,y]$,  $i$ is the smallest atom that appears as a label in $[x,y]$.  It follows that
$\lambda(x\vee j \lessdot y) = i$. We conclude that the chain $x\lessdot x\vee j \lessdot y$ has the word
of labels $ji$. Since joins are
unique, there is only one chain in $[x,y]$ with word of labels $ji$ and thus any minimum labeling
satisfies the rank two switching property.
\end{proof}

We have the following theorem as a corollary.

\begin{theorem}[\cite{DleonHallam2017}]
Every geometric lattice has a Whitney dual.
\end{theorem}

\begin{remark}For the poset $\Pi_n$ the authors proved in \cite{DleonHallam2017} that $\mathcal{ISF}_n$ is the Whitney dual corresponding to the minimal labeling of Example \ref{example:labelingPIn}.
\end{remark}

\subsection{The noncrossing partition lattice}\label{section:noncrossingpartitiondual}

\begin{figure}
\begin{center}
\input{pictures/labeling_noncrossing_partitions.tikz}
\caption{Edge labeling of $\mathcal{NC}_4$.  For clarity, the edge labels are represented by  line patterns.
The (red) solid lines represent the label $1$, the (blue) dashed lines represent $2$, and the
(black) dotted lines represent $3$.}\label{fig:NC4}
\end{center}
\end{figure}
We say a partition $\pi=B_1 / B_2/\cdots /B_k$ of
$[n]$ is \emph{noncrossing} if there are no $a<b<c<d$ such that $a,c\in B_i$ and $b,d\in B_j$ for
some $i\neq j$.
For
example, $124/35/67$ is not a noncrossing partition since $2<3<4<5$ and $\{2,4\}$ and $\{3,5\}$ are
in two different blocks, but $127/36/45$ is noncrossing.  The
\emph{noncrossing partition lattice}, denoted $\mathcal{NC}_n$, is the set of noncrossing partitions
of $[n]$ ordered by refinement. As the name suggest, $\mathcal{NC}_n$ is a lattice and has many nice
combinatorial properties
(see Simions survey article~\cite{Simion1997} for more information).  $\mathcal{NC}_n$ is an induced subposet
of $\Pi_n$, but it is not a sublattice of $\Pi_n$.  Figure~\ref{fig:NC4} depicts $\mathcal{NC}_4$.

In~\cite{Stanley1997}, Stanley found a beautiful connection between $\mathcal{NC}_n$ and a set of
combinatorial objects known as parking functions.  A \emph{parking function} of $n$ is a sequence
of $n$ positive integers $(p_1,p_2,\dots,p_n)$ with the property that when it is rearranged
in a weakly increasing order $p_{i_1}<p_{i_2}<\cdots < p_{i_n}$, then $p_{i_j}\le j$ for all $j$.
An edge labeling of $\mathcal{NC}_n$ is given in \cite{Stanley1997} with
the property that the words of labels along the maximal chains are
exactly the parking functions of $n-1$.   We will show that this edge labeling is in fact an
EW-labeling, establishing that $\mathcal{NC}_n$ has a Whitney dual.  To describe the labeling of Stanley,
first note that just as in the partition lattice $\Pi_n$, the cover relation is given by merging
two blocks
together.   Suppose that $\sigma$ is obtained from $\pi$ by merging $B_i$ and $B_j$, where $\min
B_i < \min B_j$, then define
\begin{align}\label{equation:parkingfunctionlabeling}
\lambda_{\mathcal{NC}}(\pi \lessdot \sigma)  = \max \{ a \in B_i \mid a<\min B_j\} .
\end{align}
See Figure~\ref{fig:NC4} to see this labeling for $\mathcal{NC}_4$.

The work in~\cite{Stanley1997} uses a slightly different definition of ER-labeling.
There, an ER-labeling is defined as a labeling such that each interval has a unique
weakly increasing maximal chain. It is not hard to see that the labeling in
\eqref{equation:parkingfunctionlabeling} does not fit this definition.  However, we have chosen to
define an ER-labeling as a labeling where each maximal interval has a unique strictly increasing
maximal chain.  Under this definition, one can check that $\lambda_{\mathcal{NC}}$ is indeed an ER-labeling.

In \cite{Stanley1997} this ER-labeling is used to prove that there is a local
$\mathfrak{S}_n$-action on the maximal chains of $\mathcal{NC}_{n+1}$.    This action is local in the sense that if a
transposition of the form $(i, i+1)$ acts on a maximal chain it only changes the chain in at
most the element at rank $i$.
Suppose that $[x,y]$ is an interval of rank two in $\mathcal{NC}_n$ and such that $\rho(x) =i-1$ and $\rho(y)
=i+1$.  Then for a maximal chain $\c$ in  $[x,y]$,
$$
(i,i+1) \c=
\begin{cases}
\c' & \mbox{ if $\c$ has a strict ascent or strict descent in $[x,y]$,}\\
\c & \mbox{otherwise,}
\end{cases}
$$
where $\c'$ is the unique maximal chain $[x,y]$ with the same label set as $\c$ which reverses the
labels in $\c$.  In other words, the action switches strict ascents and strict descents
and leaves equal labels fixed.   Note that this local action of $\mathfrak{S}_{n-1}$ coincides with the
action on the set of parking functions where the transposition $(i,i+1)$ permutes the letters $i$
and $i+1$ of a parking function. Under this action, there is exactly one weakly decreasing
parking function in each orbit.  The fact that this action exists immediately
implies that
Stanley's labeling of $\mathcal{NC}_n$ has the rank two switching property.  This, together with the
fact that the maximal
chains are in one-to-one correspondence with parking functions (which are all different) implies
that the labeling $\lambda_{\mathcal{NC}}$ in \eqref{equation:parkingfunctionlabeling} satisfies the
conditions of Definition
\ref{definition:EW}, and so is an EW-labeling. Hence by Theorem \ref{theorem:WimpliesWhitney} we
conclude that $\mathcal{NC}_n$ has a Whitney dual.
\begin{theorem}
The labeling $\lambda_{\mathcal{NC}}$ is an EW-labeling of $\mathcal{NC}_n$. Hence $Q_{\lambda}(\mathcal{NC}_n)$ is a Whitney dual of $\mathcal{NC}_n$.
\end{theorem}

We will now use Theorem \ref{theorem:secondcaracterizationQlambda} to provide a more familiar combinatorial description of the Whitney dual $Q_{\lambda_{\mathcal{NC}}}(\mathcal{NC}_n)$ of $\mathcal{NC}_n$.

Recall that a
Dyck path of order $n$ is a lattice path from $(0,0)$ to $(n,n)$ that never goes below the line $y=x$ and only takes steps in the directions of the vectors $(1,0)$ (East) and $(0,1)$ (North).
We will consider Dyck paths $D$ that come with a special labeling. Given an increasing sequence
$b_1<b_2<\cdots< b_{n+1}$ of positive integers, we label the point $(i-1, 0)$ of $D$ by $b_i$ (see Figure \ref{fig:labeledboard}).  In
Figure~\ref{fig:labeledDyck} we illustrate two labeled Dyck paths.

\begin{figure}

\begin{subfigure}[t]{0.5\textwidth}
\centering
\input{pictures/example_labeling_bottom_grid.tikz}
\caption{Labeling of the bottom row of the grid}\label{fig:labeledboard}
\end{subfigure}~
\begin{subfigure}[t]{0.5\textwidth}
\centering
\input{pictures/example_two_labeled_dyck_paths.tikz}
\caption{Two labeled Dyck paths}\label{fig:labeledDyck}
\end{subfigure}~
\caption{}\label{DyckPathFig}
\end{figure}

We now define a process of ``merging" two labeled Dyck paths $D_1$ and $D_2$ to obtain a new
labeled Dyck path $D$. Suppose that $D_1$ and $D_2$ have disjoint and
noncrossing label sets $B=\{b_1,b_2,\dots, b_j\}$ and $C=\{c_1,c_2,\dots, c_k\}$, where both sets
are written in increasing order and $b_1<c_1$.    Since the sets are noncrossing then there exists
an $i$ such that $b_i<c_1<c_2<\cdots < c_k<b_{i+1}$ (where we use the convention $b_{j+1}=\infty$).
Then, the new lattice path $D$, will be a path from $(0,0)$ to $(j+k, j+k)$ whose labels
along the bottom row are $b_1,b_2, \dots b_i, c_1,c_2,\dots, c_k,b_{i+1},\dots,b_j$.  From left to
right  until we reach the vertical line labeled $b_{i}$, $D$ looks exactly the same as $D_1$.
In the line labeled $b_{i}$ in $D$ we add all the north steps that $D_1$ had originally at
$b_i$ plus one additional north step followed by an additional east step from the line labeled
$b_i$
to the line labeled $c_1$. Then we glue $D_2$ where we left off in the line labeled $c_1$. After we
finish gluing $D_2$, we glue the remaining part of $D_1$ that goes from the line labeled $b_{i}$ to the line
labeled $b_j$. As an example,  suppose we wish to obtain a labeled Dyck path $D$ by merging the two labeled Dyck paths $D_1$ and $D_2$ in Figure~\ref{fig:labeledDyck} on label sets $\{1,3,6,7\}$ and $\{2,3,4\}$ respectively. We start by
creating a grid from $(0,0)$ to $(6,6)$ and label the bottom row with the (ordered) union of
the two labeled sets (see Figure~\ref{Dyck1}).  Since $1$ is the largest element in $\{1,3,5,6,7\}$
smaller than all the elements of $\{2,3,4\}$, we add in $D$ a new north step at the line labeled
$1$ and add a new east step afterwards between lines labeled $1$ and $2$.  Since at the line
labeled $1$, $D_1$ had two north steps, $D$ will have now $3$ north steps (see
Figure~\ref{Dyck2}). Next, glue $D_2$ where we left off (see
Figure~\ref{Dyck3}) and then the remaining part of $D_1$ to
obtain $D$ (see Figure~\ref{Dyck4}).

\begin{figure}
\begin{subfigure}{0.25\textwidth}
\centering

\input{pictures/dyck1.tikz}
\caption{}\label{Dyck1}

\end{subfigure}~
\begin{subfigure}{0.25\textwidth}
\centering
\input{pictures/dyck2.tikz}
\caption{}\label{Dyck2}
\end{subfigure}~
\begin{subfigure}{0.25\textwidth}

\input{pictures/dyck3.tikz}
\caption{}\label{Dyck3}
\end{subfigure}~
\begin{subfigure}{0.25\textwidth}
\input{pictures/dyck4.tikz}
\caption{}\label{Dyck4}\label{DyckPathFig2}
\end{subfigure}

\caption{The steps  required in merging the Dyck paths in Figure~\ref{DyckPathFig}.  New north and
east steps are in magenta.}
\end{figure}

In order to verify that the resulting labeled lattice path is also a
labeled Dyck path (that is, it has the same number of north and east steps and is always above the
diagonal), we rely on an equivalent definition of a Dyck path. A \emph{ballot sequence} of
length $2n$ is a $\{0,1\}$-string $s_1s_2\cdots s_{2n}$ with the same number of
$1$'s and $0$'s and such that for every $i\in [2n]$ the subword $s_1s_2\cdots s_i$ has at least as
many $1$'s as $0$'s. It is well-known that a lattice path that takes only north and east
steps is a Dyck path if and only if the sequence obtained associating to each north step a $1$
and to each east step a $0$ is a ballot sequence. Relying on this equivalent definition, we see
that in the resulting path $D$ the number of north steps and east steps is equal and the construction never breaks the property that every preamble in $D$
contains at least as many north steps as east steps. Hence $D$ is a well-defined labeled Dyck path.

Let $\mathcal{NCD}yck_n$ be the set whose objects are collections of labeled Dyck paths such that their underlying sets of labels form a noncrossing partition of $[n]$. We provide $\mathcal{NCD}yck_n$ with a partial order by defining for $F,F' \in \mathcal{NCD}yck_n$ the cover relation $F\lessdot F'$  whenever $F'$ can be
obtained from $F$ by merging exactly two of the labeled Dyck paths in $F$. Note here that each labeled Dyck path can be represented by its set of labels together with an exponent for each label. The exponent of an element $i$ being the number of north steps in the vertical line labeled $i$ in its Dyck path. This notation extends to the elements in $\mathcal{NCD}yck_n$. For example, we can denote the collection of Dyck paths in Figure~\ref{fig:labeledDyck} by $1^25^06^17^0/2^13^14^0$. In Figure \ref{fig:NCDyck} we illustrate $\mathcal{NCD}yck_4$.

\begin{figure}
\begin{center}
\input{pictures/NCDyck4.tikz}
 \caption{$\mathcal{NCD}yck_4$ (the colors follow the same standard as in Figure \ref{fig:NC4})}
    \label{fig:NCDyck}
\end{center}
\end{figure}

\begin{theorem}
For all $n\ge 1$,
$Q_{\lambda_{\mathcal{NC}}}(\mathcal{NC}_n)\cong \mathcal{NCD}yck_n$.
\end{theorem}
\begin{proof}
Theorem \ref{theorem:secondcaracterizationQlambda} characterizes
the poset $Q_{\lambda_{\mathcal{NC}}}(\mathcal{NC}_n)$ as being isomorphic to the poset $R_{\lambda_{\mathcal{NC}}}(\mathcal{NC}_n)$ whose
elements are pairs $(\pi,w)$ where $\pi \in \mathcal{NC}_n$ and $w$ is the word of labels of an ascent-free
chain in $[\hat{0},\pi]$. We show that $R_{\lambda_{\mathcal{NC}}}(\mathcal{NC}_n)\cong \mathcal{NCD}yck_n$.

Since maximal chains in $\mathcal{NC}_n$ are labeled with parking functions, when $\pi=\{[n]\}$ is the partition with a single block, we have that $w$ is a weakly decreasing parking function of length $n-1$, which are known to be in bijection with Dyck paths.  For example if $n=4$ the weakly decreasing parking functions are $(1,1,1)$,
$(2,1,1)$, $(2,2,1)$, $(3,1,1)$ and $(3,2,1)$. The bijection assigns to a parking function with $k_i$ occurrences of the label $i$ the Dyck path with $k_i$ north steps on the line $x=i-1$. In our notation, the pairs $([4],w) \in R_{\lambda_{\mathcal{NC}}}(\mathcal{NC}_4)$ can be
represented as $1^32^03^04^0$, $1^22^13^04^0$, $1^12^23^04^0$, $1^22^03^14^0$ and $1^12^13^14^0$.
Now, it is not hard to see that any interval of the form $[\hat{0},B_1/B_2/\cdots/B_k]$ is
isomorphic to the product of smaller noncrossing
partition lattices $\mathcal{NC}_{B_1}\times \mathcal{NC}_{B_2}\times \cdots \times \mathcal{NC}_{B_k}$, where $\mathcal{NC}_{B_j}$ is the
lattice of noncrossing partitions of $B_j\subset [n]$.
 Moreover the labels in any cover relation in
$[\hat{0},B_1/B_2/\cdots/B_k]$ depend only on the two blocks being merged. So any
ascent-free maximal chain can be represented as a noncrossing partition where each of the blocks
$B_j$ have been decorated with exponents representing an ascent-free maximal chain in $\mathcal{NC}_{B_j}$.
Note that words of labels on maximal chains of $\mathcal{NC}_{B_j}$ are  ``parking functions'' on $B_j$,
that is, if $B_j=\{b_1<b_2<\cdots <b_l\}$ then in the word of labels of a maximal chain the
number of occurrance of the letter $b_i$ is greater or equal to $i$ (an equivalent definition of
a parking function). For example, the chain in $[\hat{0},1457/23/6/89]$ with word of labels
$(8,4,4,2,1)$ is represented by $1^14^25^07^0/2^13^0/6^0/8^19^0$. Since $\lambda_{\mathcal{NC}}$ is an
EW-labeling, the cover relation $(\pi,w)\lessdot (\pi',w')$ in $R_{\lambda_{\mathcal{NC}}}(\mathcal{NC}_n)$ is
completely determined by the cover relation $\pi \lessdot \pi'$. Hence $(\pi',w')$ is obtained from
$(\pi,w)$ by merging two blocks $B_i$ and $B_j$ of $\pi$ such that $\min B_i < \min B_j$ and
$w'=\sort(wp)$ where $p=\lambda_{\mathcal{NC}}(\pi \lessdot \pi')=\max \{ a \in B_i \mid a<\min B_j\}$. The reader can note that this is equivalent to the definition of merging labeled Dyck paths. Indeed, in our notation this amounts to merging the weighted blocks on the sets $B_i$ and $B_j$ and increasing the exponent of $p\in B_i$ by one. For example,  if in $1^14^25^07^0/2^13^0/6^0/8^19^0$ we
merge the blocks with sets $\{1,4,5,7\}$ and $\{2,3\}$ we get $1^22^13^04^25^07^0/6^0/8^19^0$. If
we further merge the blocks with sets  $\{1,2,3,4,5,7\}$ and $\{6\}$ we get
$1^22^13^04^25^16^07^0/8^19^0$. We then have
that $R_{\lambda_{\mathcal{NC}}}(\mathcal{NC}_n)$  is indeed isomorphic to the poset $\mathcal{NCD}yck_n$.
\end{proof}

It is interesting to note the  well-known fact that the M\"obius function value of $\mathcal{NC}_n$ is (up
to a sign) the Catalan number $C_{n-1}$. This information is recovered here since $\mathcal{NCD}yck_n$ is a
Whitney dual of $\mathcal{NC}_n$ and its maximal elements are Dyck paths of order $n-1$ which  are Catalan objects.

\subsection{The weighted partition poset and the poset of rooted spanning
forests}\label{weightedPartSec} Here we discuss
the original example that motivated our work. This example was noticed by
 Gonz\'alez D'Le\'on and Wachs in \cite{DleonWachs2016} while
studying a poset of partitions where each block has a weight that is a natural number. This
poset, known as the poset of weighted partitions and denoted $\Pi_n^w$, was originally introduced
by Dotsenko and Khoroshkin in \cite{DotsenkoKhoroshkin2007} and is
related to the study of the operad of Lie algebras with two compatible brackets. The authors
of \cite{DleonWachs2016} realized that the Whitney numbers of the first and
second kind were switched with respect to those of the poset of rooted spanning forests $\mathcal{SF}_n$ on
$[n]$ studied by Reiner \cite{Reiner1978} and Sagan \cite{Sagan1983}. Since the two pairs of
Whitney numbers were already computed, by direct comparison,
we can conclude that $\Pi_n^w$ and $\mathcal{SF}_n$ are Whitney duals. In this section we use the
theory developed in Section \ref{section:whitneylabelings} to give a different proof of this fact.

A weighted partition $B_1^{v_1}/B_2^{v_2}/\cdots/B_t^{v_t}$ of $[n]$ is a partition
$B_1/B_2/\cdots/B_t$ of $[n]$ such that each block $B_i$ is assigned a weight $v_i \in
\{0,1,2,\dots,|B_i|-1\}$. The {\it poset of weighted partitions} $\Pi_n^{w}$  is the set of weighted
partitions of $[n]$ with
order relation given by
$$
A_1^{w_1}/A_2^{w_2}/\cdots/A_s^{w_t} \le B_1^{v_1}/ B_2^{v_2}/\cdots/B_t^{v_t}
$$
if the following
conditions hold:
\begin{itemize}
 \item $A_1/A_2/\cdots /A_s \le B_1/B_2/ \cdots /B_t $ in $\Pi_n$
 \item if $B_k=A_{i_1}\cup A_{i_2}\cup \cdots \cup A_{i_l} $ then
 $v_k-(w_{i_1} + w_{i_2} + \cdots + w_{i_l})\in \{0,1,\dots,l-1\}$.
\end{itemize}
Equivalently, we can define the covering relation  by
$$A_1^{w_1}/A_2^{w_2}/\dots /A_s^{w_s}\lessdot B_1^{v_1}/ B_2^{v_2}/ \dots /B_{s-1}^{v_{s-1}}$$
if the following conditions hold:
\begin{itemize}
 \item $A_1/A_2/\cdots /A_s \lessdot B_1/B_2/ \cdots /B_{s-1} $ in $\Pi_n$
 \item if $B_k=A_{i}\cup A_{j}$, where $i \ne j$, then $v_k-(w_{i} + w_{j}) \in \{0,1\}$
 \item if $B_k = A_i$ then $v_k = w_i$.
 \end{itemize}

\begin{figure}[h]

\begin{center}
\input{pictures/weighted_partition_poset_n3.tikz}
\end{center}
\caption[]{Weighted partition poset for $n=3$}\label{figure:weightedn3k2}
\end{figure}

The poset $\Pi_n^w$ has a minimum element $$\hat 0:= 1^0/2^0 / \dots /n^0$$ and  $n$
maximal elements
$$[n]^0, \, [n]^1, \dots,  [n]^{n-1}.$$ Note  that  for all   $i=0,1,\dots,n-1$, the maximal
intervals
$[\hat 0, [n]^i]$ and $[\hat 0, [n]^{n-1-i}]$ are isomorphic to each other, and the two maximal
intervals $[\hat 0, [n]^0]$ and $[\hat 0, [n]^{n-1}]$ are isomorphic to $\Pi_n$. See
Figure~\ref{figure:weightedn3k2} for the example of $\Pi_3^w$.

A \emph{rooted spanning forest} on $[n]$ is a spanning forest of the complete graph on vertex
set $[n]$  such that in every connected component there is a unique specially marked vertex, called
the
\emph{root}.
Let $\mathcal{SF}_n$ be the set of rooted spanning forests on $[n]$. For $F\in\mathcal{SF}_n$ let $E(F)$
denote the edge set of $F$ and $R(F)$ be the set of roots in $F$. The set $\mathcal{SF}_n$ has the structure
of a poset with order relation given by $F_1\le F_2$ whenever
$$E(F_1)\subseteq E(F_2) \text{ and } R(F_2) \subseteq R(F_1).$$
Equivalently, the cover relation $F_1\lessdot F_2$ occurs if $F_2$ is obtained from $F_1$
by adding a new edge $\{x,y\}\in E(F_2)$ such that $x,y \in R(F_1)$ and by choosing either $x$ or
$y$ as the new root of its component. Note that $R(F_2)$ is either $R(F_1)\setminus \{x\}$ or
$R(F_1)\setminus \{y\}$.
See Figure \ref{figure:spanning_forests_3} for the
example of $\mathcal{SF}_3$. For more on this poset see \cite{Reiner1978,Sagan1983}.

\begin{figure}[h]
\begin{center}
\input{pictures/spanning_forests_n3.tikz}
\end{center}
\caption[]{$\mathcal{SF}_3$ (here the roots are marked with red squares)}\label{figure:spanning_forests_3}
\end{figure}

Let $F \in \mathcal{SF}_n$ and $x,y$ be vertices in $F$, we say that $x$ is the \emph{parent}
of $y$ (and $y$ the \emph{child} of $x$) if $\{x,y\}$ is an edge of $F$ that belongs to the unique path from $y$ to the root of their component in $F$. We  denote $x=p(y)$ whenever $x$ is the parent of $y$. We say that
$x$
is an \emph{ancestor} of $y$ ($y$ is a descendant of $x$) if $x$  is in the unique path from $y$ to
the root of their connected component in $F$. To every forest $F\in \mathcal{SF}_n$, we can associate an
element  of $\Pi_n^w$,
\begin{align}\label{definition:piofanF}\pi(F):=\{V(T)^{w_T}\mid T
\text{ a tree in } F\},
\end{align}
where $w_T$ is the number of \emph{descents} in $T$, i.e., edges
$\{x,p(x)\}$ in $T$ where $p(x)>x$. As
an example, if $F$ is the forest formed by the trees $T_1$ to $T_6$ of
Figure~\ref{figure:example_unique_increasing_chain},  then
$w_{T_1}= 0$, $w_{T_2}= 0$, $w_{T_3}=
0$, $w_{T_4}= 3$, $w_{T_5}= 1$, $w_{T_6}= 1$ and thus $\pi(F) =
1689\underline{11}\,\underline{14}^{3}/23^{0}/4^0/5^0/\underline{10}\,\underline{12}^{1}
/7\underline{13}^{1}$ (where we have underlined the two digit numbers).
The following propositions were proved in \cite{DleonWachs2016} and \cite{Sagan1983}.

\begin{proposition}[\cite{DleonWachs2016}]\label{proposition:mobiusweightedpartitions}
For all $x\in \Pi_n^w$,
$$\mu([\hat{0},x])=(-1)^{\rho(x)}|\{F\in \mathcal{SF}_n \mid \pi(F)=x\}|.$$
In particular,
$$\mu([\hat{0},[n]^i])=(-1)^{n-1}|\mathcal{T}_{n,i}|,$$
where $\mathcal{T}_{n,i}$ is the set of rooted trees with $i$ descents.
\end{proposition}

\begin{proposition}[\cite{Sagan1983}]\label{proposition:mobiusrootedforests}
For all $F\in \mathcal{SF}_n$,
$$\mu([\hat{0},F])=\begin{cases}
(-1)^{\rho(F)} & \text{if $F$ is a forest in which every nonroot vertex is a leaf} \\
0 & \text{otherwise}.
  \end{cases} $$
\end{proposition}

Note that the map $\pi:\mathcal{SF}_n\rightarrow \Pi_n^w$ defined in Equation~\ree{definition:piofanF} is a bijection when we restrict the domain to the
set of forests in which every one of its nonroot vertices are leaves. Furthermore, note that a
forest $F \in \mathcal{SF}_n$ and its associated weighted partition $\pi(F)\in \Pi_n^w$ have the same rank
in their respective posets. We obtain the following theorem as a corollary of the previous two
propositions.

\begin{theorem}[Gonz\'alez D'Le\'on -
Wachs, personal communication]\label{theorem:weightedpartitions_rootedforests_whitneyduals}
We have that
\begin{align*}
 w_k(\Pi_n^w)&=|\{F\in \mathcal{SF}_n \mid \rho(F)=n-k\}|=W_k(\mathcal{SF}_n)\\
 w_k(\mathcal{SF})&=|\{x\in \Pi_n^w \mid \rho(x)=n-k\}|=W_k(\Pi_n^w).
\end{align*}
 Hence, the posets $\Pi_n^w$ and $\mathcal{SF}_n$ are Whitney duals.
\end{theorem}

\subsubsection{A CW-labeling of $\Pi_n^w$}

In the following we will use the structure of the poset $\mathcal{SF}_n$ and a surjective map from the poset
of saturated chains $C(\Pi_n^w)$ to $\mathcal{SF}_n$ to give a CW-labeling $\lambda_C$ for $\Pi_n^w$.
Theorem \ref{theorem:CW}  will then imply that $Q_{\lambda_C}(\Pi_n^w)$ is a Whitney dual of $\Pi_n^w$.
We will also show that $Q_{\lambda_C}(\Pi_n^w)\simeq \mathcal{SF}_n$ providing a different proof of
Theorem \ref{theorem:weightedpartitions_rootedforests_whitneyduals}.

We  first discuss some structural properties of $\mathcal{SF}_n$. For a rooted tree $T$ on vertex
set $V(T)$, let $r(T)$ denote the \emph{root} of $T$; and  for any pair of vertices $v,w\in V(T)$
define the \emph{distance} $d(v,w)$ to be the number of edges in the unique path between $v$ and
$w$. Note that by the definition of the cover relations in $\mathcal{SF}_n$, a saturated
chain $\hat{0}=F_0\lessdot F_1 \lessdot \cdots \lessdot F_k=F$ in $\mathcal{SF}_n$ can be seen as a
step-by-step instruction set on how to build the forest $F$. Here in the $i$-th step, exactly two
trees $T_{i,1}$ and $T_{i,2}$ of $F_{i-1}$ are combined by adding the edge $\{r(T_{i,1}),r(T_{i,2})\}$ to get a new tree $T'_i$ in $F_i$ whose root $r(T'_i)\in
\{r(T_{i,1}),r(T_{i,2})\}$. We call this process \emph{merging} the trees $T_{i,1}$ and $T_{i,2}$ by the roots  $r(T_{i,1})$ and  $r(T_{i,2})$. At $F_0=\hat{0}$ every element of $[n]$ is a root ($R(F)=[n]$), but at
each step $i$ there is exactly one element $v_i \in \{r(T_{i,1}),r(T_{i,2})\}$ that stops being a
root, i.e., $v_i \in R(F_{i-1})$ and $v_i \notin R(F_i)$. This process defines an ordered listing
$v_1,v_2,\dots,v_k$ of the non-root vertices of $F$.

We can also consider the converse situation:
let $v_1,v_2,\dots,v_k$ be an ordered listing of the non-root vertices of $F$ and let
$\hat{0}=F_0, F_1, \cdots ,F_k=F$ be the sequence defined by obtaining $F_{i}$  from
$F_{i-1}$ by adding the edge $\{v_i,p(v_i)\}$ (where $p(v_i)$ is the parent of $v_i$ in $F$) and letting $R(F_i)=R(F_{i-1})\setminus\{v_i\}$. It
is
clear that the rank $\rho(F_i)=i$ in $\mathcal{SF}_n$, but it is not clear if the set $\hat{0}=F_0, F_1,
\cdots ,F_k=F$ forms a saturated chain in $\mathcal{SF}_n$ since a cover relation in $\mathcal{SF}_n$ happens {\bf
exactly} when the two trees in $F_{i-1}$ are merged using an edge between their roots. The
following lemma characterizes which
sequences
$v_1,v_2,\dots,v_k$ give a valid saturated chain $\hat{0}=F_0\lessdot F_1 \lessdot \cdots \lessdot
F_k=F$ in $\mathcal{SF}_n$.

An ordered listing $v_1,v_2,\dots,v_k$  of some subset of $V(F)$ is said to be a \emph{linear
extension} if whenever $v_i$ is a descendant of $v_j$ in $F$ then we have that $i<j$.

\begin{lemma}\label{lemma:building_forests}
 The sequence $\hat{0}=F_0,F_1, \cdots , F_k=F$ in which $F_{i}$  is obtained from
$F_{i-1}$ by adding the edge $\{v_i,p(v_i)\}$ and setting $R(F_i)=R(F_{i-1})\setminus\{v_i\}$ is a
saturated chain in $\mathcal{SF}_n$ if and only if the ordered listing $v_1,v_2,\dots,v_k$ of the
non-root vertices of $F$ is a linear extension.
\end{lemma}
\begin{proof}
 First let $\hat{0}=F_0\lessdot F_1 \lessdot \cdots \lessdot F_k=F$ be a saturated chain in
$\mathcal{SF}_n$ and suppose that the associated ordered listing $v_1,v_2,\dots,v_k$ is not a linear
extension. Then there are
non-root vertices $v_i$ and $v_j$ in $F$ such that $v_i$ is a descendant of $v_j$ and $j<i$. We
choose $v_i$ and $v_j$ such that $d(v_i,v_j)$ is minimal and we claim that in this case it must be
that $v_j=p(v_i)$. Otherwise, there is a vertex $v_l$ that is a descendant of $v_j$ and an ancestor
of $v_i$. If $l < j$ then the pair $(v_i,v_l)$ satisfies the condition above with
$d(v_i,v_l)<d(v_i,v_j)$ and if $l >j$ then the pair $(v_l,v_j)$ satisfies the condition above
with
$d(v_l,v_j)<d(v_i,v_j)$. Now, if $v_j=p(v_i)$ and $j<i$ this implies that in the step $i$ between
$F_{i-1}$ and $F_{i}$ we added the edge $\{v_i,v_j\}$ but $v_j \notin R(F_{i-1})$ since $v_j$ has
been already removed from the set of roots in step $j$.  This implies that $F_i$ does not cover
$F_{i-1}$, that is a contradiction. We conclude that
$v_1,v_2,\dots,v_k$ is a linear
extension.

On the other hand, let the ordered listing $v_1,v_2,\dots,v_k$ be a linear extension of the non-root
vertices of $F$ and let $\hat{0}=F_0,F_1, \cdots , F_k=F$ in which $F_{i}$ is defined as stated in
the lemma. Note that $F_{k-1}$ is the forest obtained from $F$ by
removing the edge $\{v_k,p(v_k)\}$ and is such that $R(F_{k-1})=R(F)\cup \{v_k\}$. We also clearly
have
that $v_1,v_2,\dots,v_{k-1}$ is a linear extension of the non-root vertices of $F_{k-1}$.
Since we know that the trivial sequence $F_0=\hat{0}$ is a saturated chain in $\mathcal{SF}_n$ we assume by
induction that $\hat{0}=F_0\lessdot F_1 \lessdot \cdots \lessdot F_{k-1}$ is also one. Note that
$p(v_k)
\neq v_j$ for $j=1,\dots,k$ since the ordered listing is a linear extension and so  $p(v_k)$
cannot appear before $v_k$. But this implies that both $v_k$ and $p(v_k)$ are in $R(F_{k-1})$ and
that $F_k$ is obtained from $F_{k-1}$ by adding the edge $\{v_k,p(v_k)\}$  such that
$R(F_{k})=R(F_{k-1})\setminus \{v_k\}$.  Thus $F_{k-1}\lessdot F_k$ and so $\hat{0}=F_0\lessdot
F_1 \lessdot \cdots \lessdot F_k=F$ is a saturated chain  in $\mathcal{SF}_n$.
\end{proof}

We
define the
\emph{cost} of a rooted tree $T$ as
\begin{align}
 \gamma(T)=\sum_{v \in V(T)}d(v,r(T)).
\end{align}
See Figure \ref{figure:example_weight_of_a_tree} for an example.

\begin{figure}[h]
\begin{center}
\input{pictures/example_weight_of_a_tree.tikz}
\end{center}
\caption[]{A tree $T$ with cost $\gamma(T)=7$}\label{figure:example_weight_of_a_tree}
\end{figure}

Recall that  $C(\Pi_n^w)$ is the poset
of saturated chains from $\hat{0}$ in
$\Pi_n^w$.
For a chain $$\c:(\hat{0}=x_0\lessdot x_1 \lessdot \cdots \lessdot x_{\ell})\in C(\Pi_n^w)$$  let
$\c_i$ be the subchain of $\c$ consisting of its bottom $i+1$ elements, i.e.,
$$\c_i:(x_0\lessdot x_1 \lessdot \cdots \lessdot x_{i}).$$
We recursively define
 a map $\mathfrak{F}:C(\Pi_n^w)\rightarrow \mathcal{SF}_n$ as follows:
\begin{enumerate}
 \item Let $\mathfrak{F}(\hat{0}_{C(\Pi_n^w)})=\hat{0}_{\mathcal{SF}_n}$, the rooted forest on $[n]$ with no edges.
Note
that $\pi(\hat{0}_{\mathcal{SF}_n})=\hat{0}_{\Pi_n^w}$ where the function $\pi$ is defined in Equation~\ree{definition:piofanF}.
 \item  Let $\c:(\hat{0}=x_0\lessdot x_1
\lessdot \cdots \lessdot x_{k}) \in C(\Pi_n^w)$.
The cover relation
$$x_{k-1}:=A_1^{w_1}/A_2^{w_2}/\dots /A_s^{w_s}\lessdot B_1^{v_1}/ B_2^{v_2}/ \dots
/B_{s-1}^{v_{s-1}}=:x_k$$ in $\Pi_n^w$ is such that exactly two weighted blocks $A_i^{w_i}$
and $A_j^{w_j}$ of
$x_{k-1}$ are
merged into a weighted block $B_m^{v_m}$ of $x_k$ where  $B_m=A_{i}\cup A_{j}$ and $v_m-(w_{i}
+
w_{j}) \in \{0,1\}$.  We assume that we have recursively defined
$\mathfrak{F}(\c_{k-1})$ as a rooted spanning forest with the property that $\pi(\mathfrak{F}(\c_{k-1}))=x_{k-1}$.
We then define $\mathfrak{F}(\c)$ to be the rooted spanning forest obtained from
$\mathfrak{F}(\c_{k-1})$ by connecting the two trees $T_i$ and
$T_j$ with vertex sets $V(T_i)=A_i$ and
$V(T_j)=A_j$ using the edge $\{r(T_i),r(T_j)\}$ and choosing the root of the new tree to be
$\min\{r(T_i),r(T_j)\}$ if
$v_m-(w_{i}
+ w_{j})=0$ or   $\max\{r(T_i),r(T_j)\}$ if $v_m-(w_{i} + w_{j})=1$.
\end{enumerate}
Note that by construction and equation \eqref{definition:piofanF}, we have that $\pi(\mathfrak{F}(\c))=x_{k}$
and so the map $\mathfrak{F}$ is inductively well-defined.  Moreover, it has  the property that for every $\c \in C(\Pi_n^w)$
we have that $\pi(\mathfrak{F}(\c))=e(\c)$ (where, just as before, $e(\c)$ is maximum element of $\c$). Indeed, the
blocks of $x_k$ are the same as the ones of $x_{k-1}$ except for $A_i^{w_i}$ and
$A_j^{w_j}$ that now form the block $(A_i \cup A_j)^{v_m}$. It is not hard to see that $\mathfrak{F}$ is
sending cover relations in $C(\Pi_n^w)$ to cover relations in $\mathcal{SF}_n$ and so $\mathfrak{F}$ is a rank
preserving poset map.

\begin{lemma}\label{lemma:saturatedchainsinPIandSF}
Let $\widetilde \mathfrak{F}:C(\Pi_n^w)\rightarrow C(\mathcal{SF}_n)$ be defined for $\c:(\hat{0}=x_0\lessdot
x_1\lessdot \cdots \lessdot x_{k}) \in C(\Pi_n^w)$ by
$$\widetilde \mathfrak{F}(\c)=\mathfrak{F}(\c_0)\lessdot \mathfrak{F}(\c_1)\lessdot \cdots \lessdot \mathfrak{F}(\c_k),$$
and $\widetilde \pi:C(\mathcal{SF}_n)\rightarrow C(\Pi_n^w)$ defined for $\mathbf{F}:(\hat{0}=F_0\lessdot
F_1\lessdot \cdots \lessdot F_k)\in C(\mathcal{SF}_n)$ by
$$\widetilde \pi(\mathbf{F})=(\pi(F_0)\lessdot \pi(F_{1})\lessdot
\cdots \lessdot \pi(F_{k})).$$
Then we have that these maps are well-defined and that $\widetilde \pi \circ
\widetilde \mathfrak{F}=id_{C(\Pi_n^w)}$ and $\widetilde \mathfrak{F}\circ \widetilde \pi=id_{C(\mathcal{SF}_n)}$. Hence,
$\widetilde \mathfrak{F}$ and $\widetilde \pi$ define an isomorphism $C(\Pi_n^w)\simeq C(\mathcal{SF}_n)$.
\end{lemma}
\begin{proof}
Note that if $F\lessdot F'$, we have that $F'$ is obtained from $F$ by merging two trees $T_1$ and
$T_2$ to get a tree $T'$ of $F'$ such that $r(T')=r(T_1)$. Hence we have that $\pi(F)$ and $\pi(F')$
are almost identical except that they differ in two weighted blocks $V(T_1)^{w_{T_1}}$ and
$V(T_2)^{w_{T_2}}$
of $\pi(F)$ and one weighted block $(V(T_1)\cup V(T_2))^{w_{T_1}+w_{T_2}+\chi(r(T_1)>r(T_2))}$ of
$\pi(F')$ (where $\chi(A)=1$ if the statement $A$ is satisfied and $0$ otherwise). This means
exactly that $\pi(F)\lessdot \pi(F')$ so $\widetilde \pi$ is a well-defined map that preserves
cover relations and hence is order-preserving. The comments preceding this lemma also
imply that $\widetilde \mathfrak{F}$ is a well-defined order-preserving map.

By the recursive definition of $\mathfrak{F}$ it follows that $\widetilde \pi \circ \widetilde
\mathfrak{F}=id_{C(\Pi_n^w)}$. The reader can verify using induction on the length $k$ of a chain
$\mathbf{F}:(\hat{0}=F_0\lessdot F_1\lessdot \cdots \lessdot F_k)\in C(\mathcal{SF}_n)$ that $\widetilde \mathfrak{F}\circ
\widetilde \pi=id_{C(\mathcal{SF}_n)}$. Hence we obtain the desired isomorphism $C(\Pi_n^w)\simeq C(\mathcal{SF}_n)$.
\end{proof}
\begin{remark}
Note that Lemma \ref{lemma:saturatedchainsinPIandSF} implies that the information encoded in the
saturated chain $\c:(\hat{0}=x_0\lessdot x_1\lessdot \cdots \lessdot x_{k}) \in C(\Pi_n^w)$ can be
recovered uniquely from the saturated chain
$$\widetilde \mathfrak{F}(\c)=\mathfrak{F}(\c_0)\lessdot \mathfrak{F}(\c_1)\lessdot \cdots \lessdot \mathfrak{F}(\c_k)  \in C(\mathcal{SF}_n).$$
We will use this fact in the construction of a C-labeling of $\Pi_n^w$.
\end{remark}

We define first an E-labeling $\lambda_{\mathcal{SF}}:\mathcal{E}(\mathcal{SF}_n)\rightarrow \Lambda$ as follows: Let
$F\lessdot F'$ be such that $T_i$ and $T_j$ are the trees of $F$ that have been merged to get a
tree $T'$ of $F'$ and assume without loss of generality that $r(T_i)<r(T_j)$. We define
\begin{align}\label{equation:Elabelingforests}
\lambda_{\mathcal{SF}}(F\lessdot F')=
\begin{cases}
\left(-\gamma(T_j),r(T_i),r(T_j)\right) & \text{if $r(T')=r(T_i)$} \\
\left(-\gamma(T_i),r(T_j),r(T_i)\right) & \text{if $r(T')=r(T_j)$},
  \end{cases}
\end{align}
and we define $\Lambda$ to be the poset $\mathbb{Z}^3$ with lexicographic order.

We will define now a C-labeling $\lambda_C:\mathcal{M}\mathcal{E}(\Pi_n^w)\rightarrow \Lambda$ as follows:
Let $\c:(\hat{0}=x_0\lessdot x_1 \lessdot \cdots \lessdot x_{\rho(\Pi_n^w)})$ be a maximal chain
of $\Pi_n^w$. We define
\begin{align}\label{equation:Clabelingweightedpartitions}
\lambda_C(\c, x_{i-1} \lessdot x_i)= \lambda_{\mathcal{SF}}(\mathfrak{F}(\c_{i-1})\lessdot \mathfrak{F}(\c_i)).
\end{align}
The bottom to top
construction
that  we have used to define $\lambda_C$, i.e.,  using the information on the saturated chains from
$\hat{0}$ to $x_{i-1}$, guarantees that this labeling is a C-labeling of $\Pi_n^w$. Indeed, for any
maximal chain $\tilde \c$ that coincides with $\c$ in the bottom $d$
elements we have that $\mathfrak{F}(\tilde \c_i)=\mathfrak{F}(\c_i)$ for $i=0,1,\dots,d-1$ and so $\tilde \c$ shares
the same labels with $\c$ along the first $d-1$ edges. We will prove that $\lambda_C$ is in fact a
CW-labeling.

Note that the definition of $\lambda_C$ says that for a chain
$$\c:( \hat{0}=x_0\lessdot x_1 \lessdot \cdots \lessdot x_k),$$
the label $\lambda_C(\c, x_{i-1} \lessdot x_{i})$
only depends on the forests $\mathfrak{F}(\c_{i-1})$ and $\mathfrak{F}(\c_i)$.
Hence the label sequence
$$\lambda_C(\c, x_i \lessdot x_{i+1}), \lambda_C(\c, x_{i+1} \lessdot x_{i+2}), \dots ,
\lambda_C(\c, x_{j-1}\lessdot x_{j})$$
only depends on the sequence
$\mathfrak{F}(\c_{i})\lessdot \mathfrak{F}(\c_{i+1})\lessdot \cdots \lessdot \mathfrak{F}(\c_{j})$.
\begin{lemma}\label{lemma:increasingmergingoftrees}
Let the sequence  $F_i\lessdot F_{i+1}\lessdot \cdots \lessdot F_{j}$ be a saturated chain in $\mathcal{SF}_n$ such
that the following holds:
\begin{itemize}
 \item The forest $F_{i+1}$ is obtained from $F_{i}$ by merging two trees $T_1$ and $T_2$ of $F_{i}$
to obtain a new tree $T'$ in $F_{i+1}$ with $r(T')=r(T_1)$, and
 \item the forest $F_{j}$ is obtained by merging $T'$ with another tree $T_3$ to obtain a new
tree $T''$ with root $r(T'')=r(T_3)\neq r(T')=r(T_1)$.
\end{itemize}
Then
$$\lambda_{\mathcal{SF}}(F_{i}\lessdot F_{i+1})>\lambda_{\mathcal{SF}}(F_{j-1}\lessdot F_{j}).$$
\end{lemma}

\begin{proof}
Under these assumptions we have that $$\lambda_{\mathcal{SF}}(F_{i}\lessdot
F_{i+1})=(-\gamma(T_2),r(T_1),r(T_2))>(-\gamma(T'),r(T_3),r(T'))=\lambda_{\mathcal{SF}}(F_{j-1}\lessdot
F_{j}),$$
since $T_2$ is a proper subtree of $T'$ and so $\gamma(T')>\gamma(T_2)$.
\end{proof}

\begin{proposition}\label{proposition:weightedpartitionsCR}
 $\lambda_C:\mathcal{M}\mathcal{E}(\Pi_n^w)\rightarrow \Lambda$ is a CR-labeling.
\end{proposition}
\begin{proof}
To show that $\lambda_C$ is a $CR$-labeling we have to show that in each rooted interval
$[x,y]_{\c}$ there is a unique increasing chain. Another way to describe this is to say
that for any saturated chain $\c:(\hat{0}=x_0\lessdot x_1 \lessdot \cdots \lessdot x_k=x)$ and
 $y\in \Pi_n^w$ such that $x< y$ there is a unique saturated chain

$$\tilde \c:(\hat{0}=x_0\lessdot x_1 \lessdot \cdots \lessdot x_k\lessdot x_{k+1}\lessdot \cdots
\lessdot x_{\rho(y)}=y)$$ such that

\begin{align}\lambda_C(\tilde \c, x_k \lessdot x_{k+1})<\lambda_C(\tilde \c, x_{k+1} \lessdot
x_{k+2})<\cdots < \lambda_C(\tilde \c, x_{\rho(y)-1}\lessdot
x_{\rho(y)}).\label{equation:increasing_condition}
\end{align}

Note that by the comments that precede Lemma \ref{lemma:increasingmergingoftrees} this label
sequence only depends on the saturated chain
$\mathfrak{F}(\tilde \c_{k})\lessdot \mathfrak{F}(\tilde \c_{k+1})\lessdot \cdots \lessdot \mathfrak{F}(\tilde \c_{\rho(y)})$ in
$\mathcal{SF}_n$. Hence if we are able to determine that there is a unique saturated chain $\mathfrak{F}(\tilde
\c_{k})=F_k \lessdot F_{k+1} \lessdot \cdots \lessdot F_{\rho(y)}$ in $\mathcal{SF}_n$ with $\pi(
F_{\rho(y)})=y$ that has a sequence of labels that is increasing, then as a consequence of Lemma
\ref{lemma:saturatedchainsinPIandSF}, the chain
$$\hat{0}=x_0\lessdot x_1 \lessdot \cdots \lessdot x_k=x=\pi(F_k)\lessdot \pi(F_{k+1})\lessdot
\cdots \lessdot \pi(F_{\rho(y)})=y$$ is the unique saturated chain $\tilde \c$ with the
desired property.

 Lemma \ref{lemma:increasingmergingoftrees} implies that for $\tilde \c$ to satisfy the increasing
condition in equation \eqref{equation:increasing_condition} from steps $k+1$ to $\rho(y)$ we can
only merge trees in a way that after a root $r(T_1)$ has been chosen in a step between $k+1$
and $\rho(y)$ the same root has to continue being a root in all consecutive steps.
Now, recall from the recursive definition of the map $\mathfrak{F}$ that the process of selecting a root
depends on the value of $u \in \{0,1\}$ where $A_1^{w_1}$
and $A_2^{w_2}$ are the blocks of $x_{s-1}$ that are merged to obtain the block $(A_1\cup
A_2)^{w_1+w_2+u}$ of $x_{s}$. By the definition of the order relation in $\Pi_n^w$, we have that
each block in $y$ is of the form $(A_{1}\cup A_{2}\cup \cdots \cup A_{l})^{w_1+w_2+\cdots +
w_l+v}$ where $v \in \{0,1,\dots,l-1\}$ and $A_1^{w_1}, A_2^{w_2},\dots,A_l^{w_l}$ are blocks
of $x_{k}$. Assume that in $\mathfrak{F}(\c_k)=F_k$ the trees corresponding to these weighted blocks are
$T_1,
T_2,\dots,T_l$ and without loss of generality assume that the indexing is such that
$r(T_1)<r(T_2)<\cdots < r(T_l)$. The reader can easily check that there is a unique tree
$T'$ obtained by merging the $l$ trees by the roots step by step
selecting at each step the same root $r(T_j)$ such that exactly $v$ of the
other $l-1$ roots are
smaller than $r(T_j)$. In fact this tree is the one where $j=v+1$. All labels that come from
the step-by-step merging process that creates $T'$ are then of the form
$(-\gamma(T_i),r(T_{v+1}),r(T_i))$ for $i\in \{1,\dots,l\}\setminus\{v+1\}$. If there are two trees
$T_i$ and $T_j$ for $i,j\in \{1,\dots,l\}\setminus\{v+1\}$ such that $\gamma(T_i)<\gamma(T_j)$ then
$$(-\gamma(T_j),r(T_{v+1}),r(T_j))<(-\gamma(T_i),r(T_{v+1}),r(T_i)),$$ and if
$\gamma(T_i)=\gamma(T_j)$ but
$r(T_i)<r(T_j)$ then $$(-\gamma(T_i),r(T_{v+1}),r(T_i))<(-\gamma(T_j),r(T_{v+1}),r(T_j)).$$
Hence there is a unique increasing way of constructing $T'$ by attaching the roots $r(T_i)$ to the
selected root $r(T_{v+1})$ by going first in reverse order of $\gamma(T_i)$ and then in order of
$r(T_i)$.

As an example of the argument above, consider the forest formed by the trees $T_1$ to $T_6$ of
Figure \ref{figure:example_unique_increasing_chain} with
$r(T_1)<r(T_2)<r(T_3)<r(T_4)<r(T_5)<r(T_6)$. Suppose that we want to find an increasing maximal
chain in the rooted interval
$$\left[1689\underline{11}\,\underline{14}^{3}/23^{0}/4^0/5^0/\underline{10}\,\underline{12}^{1}
/7\underline{13}^{1},
123456789\,\underline{10}\,\underline{11}\,\underline{12}\,\underline{13}\,\underline{14}^{
7}\right]_{\c}$$ where $\c$ is a saturated chain from $\hat{0}$ to
$1689\underline{11}\,\underline{14}^{3}/23^{0}/4^0/5^0/\underline{10}\,\underline{12}^{1}
/7\underline{13}^{1}$ such that $\mathfrak{F}(\c)$ is the forest $\{T_1,T_2,T_3,T_4,T_5,T_6\}$. Since
$7-w_{T_1}-w_{T_2}-w_{T_3}-w_{T_4}-w_{T_5}-w_{T_6}=2$, the unique increasing chain $\tilde \c$ in
this rooted interval produces a tree $T'$ whose root $r(T')=r(T_3)$ (since the edges
$\{r(T_3),r(T_1)\}$ and $\{r(T_3),r(T_2)\}$ will create exactly the $2$ additional descents in
$T'$). Since $\gamma(T_4)>\gamma(T_1)=\gamma(T_5)=\gamma(T_6)>\gamma(T_2)$ the new steps in $\tilde
\c$ consist in adding first the edge $\{r(T_3),r(T_4)\}$ then the edges $\{r(T_3),r(T_1)\}$,
$\{r(T_3),r(T_5)\}$, $\{r(T_3),r(T_6)\}$ in increasing order of their roots and finally the edge
$\{r(T_3),r(T_2)\}$. This will give the sequence of labels
$$(-7,5,6)<(-1,5,2)<(-1,5,12)<(-1,5,13)<(0,5,4).$$

\begin{figure}
 \input{pictures/example_unique_increasing_chain.tikz}
 \caption{Example of the unique tree that can be constructed along an increasing chain in a rooted
interval $[x,y]_{\c}$ of $\Pi_n^w$. Here the edges have been distinguished using red for descending edges and  blue for ascending edges.}
 \label{figure:example_unique_increasing_chain}
\end{figure}

Finally, the sequence $\mathfrak{F}(\tilde \c_{k})=F_k\lessdot F_{k+1}\lessdot \cdots \lessdot
F_{\rho(y)}$ describes a process of merging the trees in $F_k$ until we obtain
the various trees in $F_{\rho(y)}$. By the discussion above we have that each tree $T'
\in F_{\rho(y)}$ uniquely determines the subsequence of steps to build it given that the
labels
$(-\gamma(T_i),r(T_{v+1}),r(T_i))$ are required to satisfy equation
\eqref{equation:increasing_condition}. Furthermore, since these labels are all distinct and belong
to the total order $\Lambda=\mathbb{Z}^3$,  there is a unique way of organizing the labels (the unique increasing
shuffle of all the increasing subsequences of labels for all trees) for the various trees
$T'$ in an increasing order, hence there is a unique sequence $\mathfrak{F}(\tilde \c_{k})=F_k\lessdot
F_{k+1}\lessdot \cdots \lessdot
F_{\rho(y)}$ with $\pi(F_{\rho(y)})=y$, that satisfy the increasing condition in equation
\eqref{equation:increasing_condition}.
\end{proof}

\begin{lemma}\label{lemma:ranktwoswitchingweightedpartitions}
 The labeling $\lambda_C$ satisfies the rank two switching property. Moreover for
every maximal chain $\c$ in $\Pi_n^w$ and $i=1,...,\rho(\Pi_n^w)$ we have that $\mathfrak{F}(\c)=\mathfrak{F}(U_i(\c))$, where $U_i(\c)$ is the unique saturated chain obtained after possibly applying a quadratic exchange to $\c$ at rank $i$.
\end{lemma}
\begin{proof}
Any rank two increasing sequence
$$\lambda_C(\c,x_{k-1}\lessdot x_k)<\lambda_C(\c,x_k\lessdot x_{k+1})$$ is determined by a
sequence $\mathfrak{F}(\c_{k-1})\lessdot \mathfrak{F}(\c_{k})\lessdot \mathfrak{F}(\c_{k+1})$ of forests. In any such sequence we
have two possible cases.
\begin{enumerate}
 \item[{\bf Case I}:] We start with trees $T_1,T_2,T_3,T_4$ of $\mathfrak{F}(\c_{k-1})$ and obtain trees
$T'$
and $T''$ of $\mathfrak{F}(\c_{k+1})$ such that $T'$ is obtained by merging $T_1$ and $T_2$ with
$r(T')=r(T_1)$; and $T''$ is obtained by merging $T_3$ and $T_4$ with $r(T'')=r(T_3)$. In this
case,
the label sequence is $(-\gamma(T_2),r(T_1),r(T_2))<(-\gamma(T_4),r(T_3),r(T_4))$ and there is
another
forest $\mathfrak{F}(\c_{k-1})\lessdot F \lessdot \mathfrak{F}(\c_{k+1})$ obtained by merging first $T_3$ and
$T_4$ and then merging $T_1$ and $T_2$ in $F$ to get $\mathfrak{F}(\c_{k+1})$. Let $\tilde
x_k=\pi( F)$ and observe (by considering the definition of the cover relations in $\Pi_n^w$) that
$\pi(\mathfrak{F}(\c_{k-1}))\lessdot \tilde x_k \lessdot \pi(\mathfrak{F}(\c_{k+1}))$. Then the chain $\tilde \c= \c \cup
\{\tilde x_k\} \setminus  \{x_k\}$ satisfies that $\lambda_C(\tilde\c,x_{k-1}\lessdot \tilde
x_k)=(-\gamma(T_4),r(T_3),r(T_4))=\lambda_C(\c,x_{k}\lessdot x_{k+1})$ and
$\lambda_C(\tilde\c, \tilde x_{k}\lessdot
x_{k+1})=(-\gamma(T_2),r(T_1),r(T_2))=\lambda_C(\c,x_{k-1}\lessdot x_{k}).$

 \item[{\bf Case II}:]
In this case we start with trees $T_1,T_2,T_3$ of $\mathfrak{F}(\c_{k-1})$ and obtain a tree $T''$
in $\mathfrak{F}(\c_{k+1})$ by first merging $T_1$ and $T_2$ to get $T'$ with $r(T')=r(T_1)$ in $\mathfrak{F}(\c_{k})$
and then merge $T'$ and $T_3$ in $\mathfrak{F}(\c_{k})$ to get $T''$. Note that by Lemma
\ref{lemma:increasingmergingoftrees}, it is necessarily true that $r(T'')=r(T')=r(T_1)$,
otherwise the labels would not be increasing.  In this case,
the label sequence is $(-\gamma(T_2),r(T_1),r(T_2))<(-\gamma(T_3),r(T_1),r(T_3))$.
Again here there is
another
forest $\mathfrak{F}(\c_{k-1})\lessdot F \lessdot \mathfrak{F}(\c_{k+1})$ obtained by merging first $T_1$ and
$T_3$ to get a tree $\tilde T'$ with $r(\tilde T')=r(T_1)$ in $ F$  and then merging $\tilde
T'$ and $T_2$ in $ F$ to get again the same $T''$ in $\mathfrak{F}(\c_{k+1})$. We let $\tilde
x_k=\pi( F)$ and observe here again that
$\pi(\mathfrak{F}(\c_{k-1}))\lessdot \tilde x_k \lessdot \pi(\mathfrak{F}(\c_{k+1}))$. Then $\tilde \c= \c \cup
\{\tilde x_k\} \setminus  \{x_k\}$ satisfies that $\lambda_C(\tilde\c,x_{k-1}\lessdot \tilde
x_k)=(-\gamma(T_3),r(T_1),r(T_3))=\lambda_C(\c,x_{k}\lessdot x_{k+1})$ and
$\lambda_C(\tilde\c, \tilde x_{k}\lessdot
x_{k+1})=(-\gamma(T_2),r(T_1),r(T_2))=\lambda_C(\c,x_{k-1}\lessdot x_{k}).$
\end{enumerate}
Note that in the two cases above the choice of the element $\tilde x_k$ is unique. A
label of the form $(-\gamma(T_2),r(T_1),r(T_2))$ determines exactly that the trees of
$\mathfrak{F}(\c_{k-1})$ with roots $r(T_1)$ an $r(T_2)$ are being merged into a tree $T'$ of $\mathfrak{F}(\c_{k})$
with root $r(T')=r(T_1)$. Hence Lemma \ref{lemma:saturatedchainsinPIandSF} implies the uniqueness
of the resulting saturated chain in $\Pi_n^w$.  Note that a common feature of the two cases above
is that after the quadratic exchange we have $\mathfrak{F}(\tilde \c_{k+1})=\mathfrak{F}(\c_{k+1})$.
Hence applying a quadratic exchange at level $k$ does not affect the sequence
$$\mathfrak{F}(\tilde \c_{k+1})\lessdot \mathfrak{F}(\tilde \c_{k+2})\lessdot \cdots \lessdot \mathfrak{F}(\tilde
\c_{\rho(\Pi_n^w)}),$$
nor the labels $\lambda_C(\c,x_{i-1}\lessdot x_i)$ for $i=k+2,\dots,\rho(\Pi_n^w)$ since they
depend only on this sequence. In particular we have $\mathfrak{F}(\c)=\mathfrak{F}(\tilde \c)=\mathfrak{F}(U_i(\c))$. Also,
the choice of the element $\tilde x_k$ in the two cases above only depends on $\mathfrak{F}(\c_{k+1})$ and so
it has to be the same for any other chain that coincides with $\c$ in the bottom $k+2$ elements. We
conclude then that $\lambda_C$ satisfies the rank two switching property.
\end{proof}

We note here that whenever $\c \in C(\Pi_n^w)$, we have $\widetilde \mathfrak{F}(\c)=(\mathfrak{F}( \c_{0})\lessdot
\mathfrak{F}(\c_{1})\lessdot \cdots \lessdot \mathfrak{F}(\c_{k}))$ in $C(\mathcal{SF}_n)$. Since the sequence of ordered
pairs
$(r(T_{i,1}),r(T_{i,2}))$ of roots that are being merged at each step $i$  (selecting $r(T_{i,1})$
as the new root) provides enough
information to reconstruct the element of $C(\mathcal{SF}_n)$, by Lemma
\ref{lemma:saturatedchainsinPIandSF}, it also provides enough information to reconstruct the
elements in $ C(\Pi_n^w)$. Hence the sequences of labels given by $\lambda_C$
uniquely determine elements in $ C(\Pi_n^w)$. This also implies that the same is true for
all maximal chains in rooted intervals $[x,y]_{\c}$, or otherwise we can extend two maximal chains
$\c_1$ and $\c_2$ with the same word of labels to $\c \cup \c_1$ and $\c\cup \c_2$ with the same
property. This together with Proposition \ref{proposition:weightedpartitionsCR}, Lemma
\ref{lemma:ranktwoswitchingweightedpartitions} and Theorem \ref{theorem:CW} imply
the following corollary.
\begin{corollary}
 $\lambda_C$ is a CW-labeling and hence $Q_{\lambda_C}(\Pi_n^w)$ is a Whitney dual of $\Pi_n^w$
\end{corollary}

Note here that Lemma \ref{lemma:ranktwoswitchingweightedpartitions} also implies  that the map
$\mathfrak{F}:C(\Pi_n^w)\rightarrow \mathcal{SF}_n$ has the property that if $\c$ and $\c'$ are such that $[\c]=[\c']$
in $Q_{\lambda_C}(\Pi_n^w)$ then $\mathfrak{F}(\c)=\mathfrak{F}(\c')$. Hence we obtain a well-defined map
$\overline \mathfrak{F}: Q_{\lambda_C}(\Pi_n^w)\rightarrow \mathcal{SF}_n$ given by $\overline \mathfrak{F}(X)=\mathfrak{F}(\c)$ for any
$\c \in X$.

\begin{theorem}\label{theorem:QSF}
 The map $\overline \mathfrak{F}: Q_{\lambda_C}(\Pi_n^w)\rightarrow \mathcal{SF}_n$ is a poset isomorphism.
\end{theorem}
\begin{proof}
To be able to prove that this is a poset isomorphism we should show that $\overline \mathfrak{F}$ is a bijection and that
$\overline \mathfrak{F}$ and $\overline \mathfrak{F}^{-1}$ are both poset (order preserving) maps.

{\bf $\overline \mathfrak{F}$ is a bijection:} To prove this we will show that for
any $F \in \mathcal{SF}_n$ there exist a unique ascent-free chain $\c \in C(\Pi_n^w)$ such that
$\mathfrak{F}(\c)=F$. The conclusion then follows from the fact that for every  $X \in
Q_{\lambda_C}(\Pi_n^w)$ there is a unique ascent-free chain $\c \in C(\Pi_n^w)$ such that $X=[\c]$.

Recall from Lemma \ref{lemma:building_forests} that a sequence $\hat{0}=F_0\lessdot F_1
\lessdot \cdots \lessdot F_k=F$ is a saturated chain in $\mathcal{SF}_n$ if and only if the associated
ordered listing $v_1,v_2,\dots,v_k$ of the non-root vertices of $F$ is a linear
extension.
Call $T(v_i)$ the induced rooted subtree of $F$ formed by all descendants of $v_i$ (including
itself) and recall that $p(v_i)$  is the
the parent of $v_i$ with respect to $F$.
Since
$v_1,v_2,\dots,v_k$ is a linear extension,  every $v_j \in T(v_i)$ satisfies $j<i$. This
implies that every saturated chain $\hat{0}=F_0\lessdot F_1 \lessdot \cdots \lessdot F_k=F$ has
associated labels of the form $\lambda_{\mathcal{SF}}(F_{i-1}\lessdot F_i)=(-\gamma(T(v_i)),p(v_i),v_i)$.
 Hence all saturated chains
$\hat{0}=F_0\lessdot F_1 \lessdot \cdots \lessdot F_k=F$ in $[\hat{0}, F]$ for a given $F \in
\mathcal{SF}_n$ have the same set of labels
$$L(F):=\{(-\gamma(T(v_i)),p(v_i),v_i)\mid i\in [k]\}.$$
The labels in $L(F)$  are clearly all different and come from a totally ordered set, so there is a
unique ascent-free way to order
them $$(-\gamma(T(v_{i_1})),p(v_{i_1}),v_{i_1})>(-\gamma(T(v_{i_2})),p(v_{i_2}),v_{i_2})>\cdots >
(-\gamma(T(v_{i_k})),p(v_{i_k}),v_{i_k}).$$
Let $v_{i_1},v_{i_2},\dots,v_{i_k}$ be the ordered listing that we obtain in this
way.  We want to check that this sequence is also a linear extension which, by  Lemma
\ref{lemma:building_forests}, implies that it has an associated saturated chain from
$\hat{0}$ to $F$. Indeed, if this ordered listing is not a linear extension then there are vertices
$v_{i_l}$ and $v_{i_j}$ such that $v_{i_l}$ is an ancestor of $v_{i_j}$ and $l<j$. But then we have
that $(-\gamma(T(v_{i_l})),p(v_{i_l}),v_{i_l})<(-\gamma(T(v_{i_j})),p(v_{i_j}),v_{i_j})$ since
$\gamma(T(v_{i_l}))>\gamma(T(v_{i_j}))$, a contradiction.
Hence the sequence $v_{i_1},v_{i_2},\dots,v_{i_k}$ is a linear extension that gives a valid chain
$\hat{0}=F_0\lessdot F_1 \lessdot \cdots \lessdot F_k=F$ in
$\mathcal{SF}_n$  and so by Lemma \ref{lemma:saturatedchainsinPIandSF} $\c:(\hat{0}=\pi(F_0)\lessdot
\pi(F_1) \lessdot \cdots \lessdot \pi(F_k)=\pi(F))$
is the unique ascent-free chain with $\mathfrak{F}(\c)=F$.

{\bf $\overline \mathfrak{F}$ and $\overline \mathfrak{F}^{-1}$ are poset maps:} The fact that $\mathfrak{F}$ is order
preserving implies, using Definition \ref{definition:quotientposet}, Proposition
\ref{proposition:quotientposet} and the well-definedness of
$\overline \mathfrak{F}$,  that $\overline \mathfrak{F}$ is also order preserving. Now, if we have $F\lessdot F'$
in
$\mathcal{SF}_n$,
given that $\mathfrak{F}$ is surjective, there is a chain $\c$ such that $F=\mathfrak{F}(\c)$ and the recursive
definition of $\mathfrak{F}$ implies that $F'=\mathfrak{F}(\c \cup \pi(F'))$. But $\c \lessdot (\c \cup \pi(F'))$ in
$C(\Pi_n^w)$ and so $\overline \mathfrak{F}^{-1}(F)=[\c] \lessdot [(\c \cup \pi(F'))]=\overline
\mathfrak{F}^{-1}(F')$ in $Q_{\lambda_C}(\Pi_n^w)$. Since all posets are finite this implies $\overline
\mathfrak{F}^{-1}$ is order preserving.
\end{proof}

\begin{remark}
 As we mentioned at the beginning of this subsection, Theorem \ref{theorem:QSF} provides a new proof of Theorem
\ref{theorem:weightedpartitions_rootedforests_whitneyduals} as a corollary.
\end{remark}

\subsubsection{A different Whitney dual for $\Pi_n^w$}
In \cite{DleonWachs2016}, Gonz\'alez D'Le\'on and Wachs gave an ER-labeling (that is in fact an
EL-labeling) for $\Pi_n^w$, quite different from the CR-labeling constructed above.

The map $\lambda_E:\mathcal{E}(\Pi_n^w)\rightarrow \Lambda_n$ was defined as follows: let $x\lessdot y $ in
$\Pi^w_n$ so that $y$ is obtained from $x$ by merging two blocks $A^{w_A}$ and $B^{w_B}$ into a
new block $(A \cup B)^{w_A + w_B+u}$, where $u \in \{0,1\}$ and where we assume without loss of
generality that $\min A < \min B$.  We
define $$\lambda_E(x \lessdot y) = (\min A, \min B)^u.$$
Here $\Lambda_n$ is defined as follows: for each $a \in [n]$, let $\Gamma_a:= \{(a,b)^u :
a<b \le n+1, \,\, u \in \{0,1\} \}$.
We partially order $\Gamma_a$  by letting $(a,b)^u \le  (a,c)^v$ if $b\le  c$ and $u \le v$.
Note that $\Gamma_a$ is isomorphic to the direct product of the chain $a+1< a+2 <\dots < n+1 $ and
the chain $0 < 1$.  Now define $\Lambda_n$ to be the
ordinal sum
$\Lambda_n := \Gamma_1 \oplus  \Gamma_2  \oplus \cdots \oplus \Gamma_{n}$. See Figure
\ref{figure:weightedn3k2EL} for an example.

The labeling $\lambda_E$ has the property that when restricted to the intervals $[\hat 0, [n]^0]$
and $[\hat 0, [n]^{n-1}]$,  which are both isomorphic to $\Pi_n$, it reduces to the
minimal labeling of $\Pi_n$ in \cite{Bjorner1982,Stanley1974}.

\begin{figure}[h]

\begin{center}
\input{pictures/weighted_partition_poset_n3EL.tikz}
\end{center}
\caption[]{ER-labeling of  $\Pi_3^w$}\label{figure:weightedn3k2EL}
\end{figure}

\begin{theorem}[\cite{DleonWachs2016} Theorem 3.2] $\lambda_E$ is an ER-labeling.

\end{theorem}

\begin{theorem}\label{theorem:lambdaE_EW}
 $\lambda_E$ is an EW-labeling and hence $Q_{\lambda_E}(\Pi_n^w)$ is a Whitney dual of $\Pi_n^w$.
\end{theorem}
\begin{proof}Note that the information contained in the labels $(\min A, \min B)^u$ is enough to
recover any saturated chain from $\hat{0}$. Hence the sequence of labels in each interval
uniquely determines a chain. To show that $\lambda_E$ is an EW-labeling we are left to show that it
satisfies the rank two switching property.
 The rank two intervals $[x,y]$ in $\Pi_n^w$ are of three possible  different types (see Figure
\ref{figure:ranktwointervals_weighted}), we will show
that in each of these types the rank two switching property is satisfied.

\begin{enumerate}
\item[{\bf Type I}:] Two pairs of distinct blocks $\{A^{w_A},B^{w_B}\}$ and $\{C^{w_C},D^{w_D}\}$
of $x$ are merged to get $y$. Assume without loss of generality that $\min A < \min B$, $\min
A < \min C$ and $\min C < \min D$.  The open interval
$(x,y)$ equals $\{z_1,z_2\}$ where $z_1$ is like $x$, but with a block $(A\cup B)^{w_A+w_B+u_1}$
instead of $\{A^{w_A},B^{w_B}\}$ and $z_2$ is like $x$, but with a block $(C\cup D)^{w_C+w_D+u_2}$
instead of $\{C^{w_C},D^{w_D}\}$, where $u_1,u_2 \in \{0,1\}$.  In this case this interval has two
maximal chains with labels
$$\lambda_E(x\lessdot z_1)=(\min A, \min B)^{u_1}<(\min C,\min D)^{u_2}=\lambda_E(z_1\lessdot y),
\text{ and}$$
$$\lambda_E(x\lessdot z_2)=(\min C, \min D)^{u_2}>(\min A,\min B)^{u_1}=\lambda_E(z_2\lessdot y)$$

\item[{\bf Type II}:] Three distinct blocks $\{A^{w_A},B^{w_B},C^{w_C}\}$ of $x$ are merged to get
$y$, adding at each merging step the same weight $u \in \{0,1\}$.   Assume without loss of
generality that $\min A < \min B<\min C$. The open interval
$(x,y)$ equals $\{z_1,z_2,z_3\}$, where each weighted partition  $z_i$ is obtained from $x$ by
merging two of the three blocks and adding $u$ to the total weight of the resulting block.   In
this case the interval has a unique increasing chain with labels
$$\lambda_E(x\lessdot z_1)=(\min A, \min B)^{u}<(\min A,\min C)^{u}=\lambda_E(z_1\lessdot y),$$
and there is a unique chain with labels
$$\lambda_E(x\lessdot z_2)=(\min A, \min C)^{u}>(\min A,\min B)^{u}=\lambda_E(z_2\lessdot y).$$

\item[{\bf Type III}:] Three distinct blocks $\{A^{w_A},B^{w_B},C^{w_C}\}$ of $x$ are merged to
get $y$ in one step adding $0$ to the total weight and in the other step adding $1$. Assume without
loss of generality that $\min A < \min B<\min C$. The open interval
$(x,y)$ equals $\{z_1,z_2,z_3,z_4,z_5,z_6\}$, where each weighted partition  $z_i$ is obtained from
$x$ by either merging two of the three blocks and adding either $0$ or $1$ to the total weight of
the resulting block.    In
this case the interval has a unique increasing chain with labels
$$\lambda_E(x\lessdot z_1)=(\min A, \min B)^{0}<(\min A,\min C)^{1}=\lambda_E(z_1\lessdot y),$$
and there is a unique chain with labels
$$
    \lambda_E(x\lessdot z_2)=(\min A, \min C)^{1}>(\min A,\min B)^{0}=\lambda_E(z_2\lessdot y).
$$
\end{enumerate}
Thus, $\lambda_E$ has the rank two switching property.
\end{proof}

\begin{figure}
\centering
\input{pictures/types_of_intervals_weighted.tikz}
\label{fig:type1}
\caption[]{Rank two intervals in $\Pi_n^w$}\label{figure:ranktwointervals_weighted}
\end{figure}

Now that we know, by Theorem \ref{theorem:lambdaE_EW}, that $\lambda_E$ is an EW-labeling of
$\Pi_n^w$, we can  use Theorem \ref{theorem:secondcaracterizationQlambda} to describe
$Q_{\lambda_E}(\Pi_n^w)$. We leave the general characterization of $Q_{\lambda_E}(\Pi_n^w)$ for a
future article, but we explicitly compute the example of $Q_{\lambda_E}(\Pi_3^w)$ in Figure
\ref{figure:second_whitney_dual_Pi3}. An interesting fact here is that $Q_{\lambda_E}(\Pi_3^w)$
and
$Q_{\lambda_C}(\Pi_3^w)\simeq \mathcal{SF}_3$ are evidently not isomorphic (see Figures
\ref{figure:spanning_forests_3} and \ref{figure:second_whitney_dual_Pi3}).

\begin{theorem}\label{theorem:Qlambdanonisomorphic}
 There exist a poset $P$ and two CW-labelings $\lambda_1$ and $\lambda_2$ of $P$ such that the
posets $Q_{\lambda_1}(P)$ and $Q_{\lambda_2}(P)$ are not isomorphic.
\end{theorem}

\begin{figure}[h]
\begin{center}
\input{pictures/second_whitney_dual_Pi3.tikz}
\end{center}
\caption[]{$Q_{\lambda_E}(\Pi_3^w)$}\label{figure:second_whitney_dual_Pi3}
\end{figure}

\subsection{R$^*$S-labelable posets}  In~\cite{SimionStanley1999} Simion and Stanley introduced the notion
of an R$^*$S-labeling as a tool to study local actions of the symmetric group on maximal chains of
a poset. In this subsection, we show that the R$^*$S labelings that respect the consistency
condition of the rank two switching property are CW-labelings.   Simion and Stanley~\cite{SimionStanley1999} showed the posets of shuffles $W_{M,N}$ introduced by
Greene~\cite{Greene1988}
have R$^*$S labelings and Hersh~\cite{Hersh1999} showed that the noncrossing partition lattices
$NC_n^B$ and $NC_n^D$ of
types B and D introduced by Reiner~\cite{Reiner1997} also have such labelings.   As noted
in~\cite{SimionStanley1999},  Stanley's parking function labeling of the noncrossing partition
lattice $NC_n$ of type $A$ described in a previous subsection  is an R$^*$S labeling. All of these
are examples of  R$^*$S labelings that are CW-labelings, hence we have as a corollary that these
posets have Whitney duals.

We start with the definition of an S-labeling of a poset given in~\cite{SimionStanley1999}.  We
present it here in a slightly different language than how was originally stated in
\cite{SimionStanley1999} to highlight the connection with
CW-labelings.

\begin{definition} [\cite{SimionStanley1999}]\label{rStarDef}
Let $P$ be a graded poset  of rank $n$ with a $\hat{0}$ and  $\hat{1}$. Let $\lambda$ be a
C-labeling such that the labels are totally ordered. We say $\lambda$ is an S-labeling if
\begin{enumerate}
\item \label{condition:S1} For each  maximal chain $\c =  (\hat{0}=x_0\lessdot x_1 \lessdot \cdots
\lessdot x_n =\hat{1})$
and for each $1\leq i \leq n-1$ such that $\lambda(\c, x_{i-1} \lessdot x_i) \neq  \lambda(\c, x_{i}
\lessdot x_{i+1})$, there exists a unique maximal chain $\c' =   (\hat{0}=x_0\lessdot x_1 \lessdot \cdots
\lessdot x_{i-1}\lessdot x'_{i}\lessdot x_{i+1} \lessdot \cdots  \lessdot x_n =\hat{1})$ such that $\c$ and
$\c'$ have the same sequence of labels except that  $\lambda(\c, x_{i-1} \lessdot x_i )  =
\lambda(\c', x'_{i} \lessdot x_{i+1})$ and $\lambda(\c, x_{i} \lessdot x_{i+1} )  = \lambda(\c',
x_{i-1}
\lessdot x'_{i})$
\item \label{condition:S2} $\lambda$  is one-to-one on maximal chains.  That is,  two different
maximal chains must have
different sequences of labels from  bottom to top.
\end{enumerate}
\end{definition}

Considering the condition \eqref{condition:S1} of an S-labeling in Definition~\ref{rStarDef}, one
can see that
if the choice of the element $x'_{i}$ is consistent among all maximal chains that coincide in
the first $i+2$ edges then the S-labeling also satisfies the rank two
switching property of Definition \ref{definition:ranktwoC}. We will call S-labelings satisfying
this additional condition \emph{consistent}. In that case, we note that condition
\eqref{condition:S1} on S-labelings is in fact stronger than the rank two switching property.  For
example, the labeling  of $\Pi_3$ given in
Figure~\ref{figure:exampleER}  has the rank two switching property,  but it is not an S-labeling
since
the chain with label sequence $(2,3), (1,3)$ cannot be switched with anything as it
should be in the case of an S-labeling.  Also, note that condition \eqref{condition:S2} of
Definition \ref{rStarDef} is also stronger than the simpler requirement of Definition
\ref{definition:EW} that ascent-free chains are one-to-one. Hence any S-labeling satisfying the
consistency condition and which is also a CR-labeling, is a CW-labeling.

In~\cite{SimionStanley1999} Simion and Stanley refer to C-labelings that are also  CR-labelings  as
R$^*$-labelings. Moreover, any labeling which is both an R$^*$-labeling and an S-labeling is
called an R$^*$S-labeling.
We warn the reader of a possible source of confusion since in this paper
we have used the term ER$^*$ to mean a different type of labeling.   Using
Theorem~\ref{theorem:CW}, we have the following.

\begin{theorem}
A consistent R$^*$S-labeling is a CW-labeling.  Consequently, every poset with a consistent
R$^*$S-labeling has a Whitney dual.
\end{theorem}

\begin{remark}
 The consistency condition is automatically satisfied for E-labelings and so it only needs to be
checked when
the underlying labeling is a C-labeling.
\end{remark}

The reader can check in \cite{SimionStanley1999} that the labeling for the poset of shuffles
$W_{M,N}$ is an example of a consistent R$^*$S-labeling and hence a CW-labeling.
In \cite{Stanley1997} and \cite{Hersh1999} edge labelings of the noncrossing
partition lattices of type A, B and D are given. These are all examples of R$^*$S-labelings and hence they are also
EW-labelings. We then have the following corollary.

\begin{corollary}
The poset of shuffles $W_{M,N}$ and the noncrossing partition lattices of type A, B and D all have
Whitney duals.
\end{corollary}

\section{$H_n(0)$-actions and Whitney labelings}\label{section:heckeaction}
In this section we describe an action of the $0$-Hecke algebra on the maximal chains of
a poset $P$ with a generalized CW-labeling $\lambda$.  We will also see that the same action can be
associated to the Whitney dual $Q_\lambda(P)$ constructed in Section
\ref{section:whitneylabelings}.  The characteristic of this action is Ehrenborg's flag
quasisymmetric function in the case of $P$ and is Ehrenborg's flag quasisymmetric function with
$\omega$ applied in the case of $Q_\lambda(P)$.  The techniques we
describe here   closely  follow the work of McNamara in \cite{McNamara2003} who studied
actions of this kind on posets with EL-labelings in which the word of labels in every chain is
a permutation of $\mathfrak{S}_n$, also known as $\mathfrak{S}_n$ EL-shellable  or \emph{snellable} posets.

\subsection{An action of the $0$-Hecke algebra}
Suppose that $P$ is a graded poset of rank $n$.  Moreover, suppose that  $\lambda$ is a generalized
CW-labeling of $P$.   Recall that  $\mathcal{M}_P$ is the set of maximal chains of $P$.
Define maps $U_1,U_2,\dots,U_{n-1}: \mathcal{M}_P\rightarrow \mathcal{M}_P$ such that for
$\c:(\hat{0}=x_0\lessdot x_1\lessdot\cdots \lessdot x_n)$

$$
U_i (\c)=
\begin{cases}
\c' & \mbox{if $\lambda(\c,x_{i-1}\lessdot x_i)<\lambda(\c, x_{i}\lessdot x_{i+1})$,}\\
\c & \mbox{otherwise,}
\end{cases}
$$
where $\c'$ the unique maximal chain of $P$ obtained by applying a quadratic exchange at rank $i$. As an example, consider the maximal chain $\c:(1/2/3/4\lessdot 13/2/4\lessdot 123/4\lessdot 1234)$ in $\mathcal{NC}_4$ with the parking function labeling (see Figure~\ref{fig:NC4}).  Since there is no ascent at rank $1$, $U_1(\c) =\c$.  However, there is an ascent at rank $2$, and $U_2(\c)= 1/2/3/4\lessdot 13/2/4\lessdot 134/2\lessdot 1234$.

We
note that in \cite{McNamara2003}, where the labelings are snellings,
the maps $U_i$ are similar except that instead of exchanging ascents by descents, they
exchange descents by ascents.

\begin{proposition}\label{proposition:heckerelations}
The maps $U_1,U_2,\dots, U_{n-1}$ have the following properties.
\begin{enumerate}
\item For all $\c\in \mathcal{M}_P$, $U_i(\c)$ and $\c$ are the same
except possibly at rank $i$.
\item $U_i^2 = U_i$ for all $i$.
\item $U_iU_j = U_jU_i $ for all $i,j$ such that $|i-j|>1$.
\item  $U_iU_{i+1}U_i = U_{i+1}U_iU_{i+1}$ for all
$i$.
\end{enumerate}
\end{proposition}
\begin{proof}
 The first three properties are immediate from the definition of $U_i$ and Definition
\ref{definition:ranktwoC} of the rank two switching property for C-labelings.
The
last property is a consequence of the braid relation in a generalized CW-labeling when there is a
critical condition at rank $i$ and is easily verified when there is no critical condition at rank
$i$.
\end{proof}

The $0$-Hecke algebra of type $A$ is defined by abstract generators satisfying the same relations of those in Proposition \ref{proposition:heckerelations}. Thus the properties described in the proposition imply that there is an action of the generators of the $0$-Hecke algebra $H_n(0)$ on the set $\mathcal{M}_P$.  This action is said to be \emph{local} since the chains $U_i(\c)$  and $\c$ are the same except possibly at rank $i$.  Moreover, this action gives rise to a representation of the $0$-Hecke algebra on the space $\mathbb{C}\mathcal{M}_P$ linearly spanned by $\mathcal{M}_P$.

It turns out that the characteristic of this action is a well-known quasisymmetric function.
Before
we look at this characteristic, we need to review some material on quasisymmetric functions.

\subsection{Ehrenborg's flag quasisymmetric function}
In~\cite{Ehrenborg1996}, Ehrenborg introduced a formal power series now known as \emph{Ehrenborg's
flag
quasisymmetric function}.  Given a graded poset $P$ with a $\hat{0}$ and $\hat{1}$, it is defined by
$$
F_P(x_1,x_2,\dots ) = F_P(\mathbf{x}) = \sum_{\hat{0}=t_1\leq t_2\leq \cdots \leq t_{k-1}< t_k = \hat{1}}
x_{1}^{rk(t_0,t_1)}x_{2}^{rk(t_1,t_2)}\cdots x_{k}^{rk(t_{k-1},t_{k})}
$$
where the sum is over all multichains from $\hat{0}$ to $\hat{1}$ where $\hat{1}$ appears exactly
once.  As the name suggests, $F_P(\mathbf{x})$ belongs to the ring of quasisymmetric functions.  That is,
for each sequence $n_1,n_2,\dots, n_k$ the monomial $x_{i_1}^{n_1}x_{i_2}^{n_2}\cdots
x_{i_k}^{n_k}$ has the same coefficient as
$x_{j_1}^{n_1}x_{j_2}^{n_2}\cdots x_{j_k}^{n_k}$ whenever $i_1<i_2<\cdots< i_k$ and $j_1<j_2<\cdots
<j_k$.   In addition to being a quasisymmetric function, $F_P(\mathbf{x})$ also keeps track of the flag
$f$-vector and the flag $h$-vector of $P$ as we describe next.

Let $P$ be a graded poset with a $\hat{0}$ and $\hat{1}$.  For $S\subseteq [n-1]$ define
$$
\alpha_P(S) = |\{\hat{0}<x_1<x_2<\cdots <x_{|S|}<\hat{1}\mid \{\rho(x_1),\rho(x_2),\dots ,
\rho(x_{|S|})\} = S\}|.
$$
In other words, $\alpha_P(S)$ is the number of chains from $\hat{0}$ to $\hat{1}$ which
use elements whose rank set is $S$. The function given by $\alpha_P: 2^{[n-1]} \rightarrow
\mathbb{Z}$
is called the \emph{flag f-vector} of $P$.  We also define
$$
\beta_P(S) = \sum_{T\subseteq S} (-1)^{|S\setminus T|} \alpha_P(T).
$$
The function $\beta_P: 2^{[n-1]}\rightarrow \mathbb{Z}$ is called the \emph{flag h-vector} of $P$.
The reason for the names flag $f$-vector and flag $h$-vector is that they refine the classical
$f$-vector and $h$-vector of the order complex of $P$.  See~\cite{Stanley2012}[\S 3.13] for
more details.

When $P$ has a $\hat{0}$ and $\hat{1}$, there is a nice relationship between $F_P(\mathbf{x})$ and
$\beta_P(S)$.  Indeed, it is well-known that if $P$ has rank $n$, then
$$
F_P(\mathbf{x})  =  \sum_{S\subseteq [n-1]}  \beta_P(S) L_{S,n}(\mathbf{x})
$$
where $L_{S,n}$ is Gessel's fundamental quasisymmetric function defined by
$$
L_{S,n}(\mathbf{x}) = \sum_{\substack{1\leq i_1\leq i_2\leq \cdots  \leq i_n\\ i_j<i_{j+1} \mbox{ if } j\in
S}
} x_{i_1}x_{i_2}\cdots x_{i_n}.
$$

The original definition of $F_P(\mathbf{x})$ requires that $P$ have a $\hat{1}$, however we would like to extend this
to more general posets.   In order to do this, we consider a slight generalization of $F_P(\mathbf{x})$ to
deal with posets with a single minimal element $\hat{0}$, but possibly with multiple maximal elements.  Let $P$ be
a graded poset with a $\hat{0}$, then we
define
$$
F_P(\mathbf{x}) =  \sum_{m}F_{[\hat{0},m]}(\mathbf{x})
$$
where the sum is over all maximal elements $m$ of $P$. Note that in the case that $P$ has a
$\hat{1}$, this is just Ehrenborg's classical
definition.
Since  intervals always have a $\hat{0}$ and
a
$\hat{1}$, we have that
$$
F_P(\mathbf{x}) = \sum_{m} \left( \sum_{S\subseteq [n-1]} \beta_{[\hat{0},m]} (S) L_{S,n} \right).
$$

Now suppose that $\lambda$ is a CR-labeling of $P$ and  that $P$ has a $\hat{0}$ and a $\hat{1}$.
Recall that $\mathcal{M}_P$ denotes the set of maximal chains in $P$.  For $\c: (x_0\lessdot
x_1\lessdot
x_2\lessdot \cdots \lessdot x_n)\in\mathcal{M}_P$,  the \emph{descent set} of $\c$ is defined
to be
$$
D(\c) = \{i \mid \lambda(\c, x_{i-1},x_i) \nless \lambda(\c, x_i,x_{i+1})\}.
$$
It was shown by Stanley~\cite{Stanley2012}[c.f. Theorem 3.14.2] for ER-labelings and of Bj\"orner
and Wachs \cite{BjornerWachs1997} for CR-labelings that $\beta_S(P)$ is  the number of
maximal chains with
descent set  $S$.
A simple modification of Stanley's proof of the combinatorial description of the numbers $\beta_P(S)$, shows that if $\lambda$ is an ER$^*$-labeling of a poset $P$ with $\hat{0}$ and $\hat{1}$, then $\beta_{P}(S) =\{\c \in
\mathcal{M}_P\mid D(\c) =S^c\}$ for any $S\subseteq [n-1]$.

\begin{example}
We compute $F_P(\mathbf{x})$ for $P=\Pi_3$. As one  can see in
Figure~\ref{figure:exampleER},
$\Pi_3$
has three maximal chains.  Under the ER-labeling of Example \ref{example:labelingposetofpartitions}, one of the maximal chains is increasing  and the other two are ascent-free.   It
follows that $\beta_{\Pi_3}(\emptyset) = 1$ and $\beta_{\Pi_3}(\{1\}) = 2$.  Thus,
$$
F_{\Pi_3}(\mathbf{x}) = L_{\emptyset, 2}(\mathbf{x})+2L_{\{1\},2}(\mathbf{x}).
$$
\end{example}

\begin{example}
We consider the example of $F_P(\mathbf{x})$ when $P = \mathcal{ISF}_3$.  As  one can see in
Figure~\ref{figure:exampleER},
there are two maximal intervals with three maximal chains altogether.  Moreover, using the ER$^*$-labeling of Example \ref{example:labelingISF} we can compute  $\beta_{[\hat{0},m]}(S)$ with the number of maximal chains with strict ascent set given by $S$.

If $F_1$
is the increasing spanning forest with edge set $\{(1,2), (1,3)\}$ and $F_2$ is the one with
$\{(1,2), (2,3)\}$, we see that
$$
\beta_{[\hat{0},F_1]}(\emptyset) = 1,  \beta_{[\hat{0},F_1]}(\{1\})
=1,\beta_{[\hat{0},F_2]}(\emptyset) =1, \mbox{and }
\beta_{[\hat{0},F_2]}(\{1\}) =0.
$$
Therefore
$$
F_{\mathcal{ISF}_3}(\mathbf{x})  = 2 L_{\emptyset, 2}(\mathbf{x})  +L_{\{1\},2}(\mathbf{x}).
$$
\end{example}

\begin{example}
We compute $F_P(\mathbf{x})$ for $P=\mathcal{NC}_4$. Recall from Section \ref{section:noncrossingpartitiondual} that $\mathcal{NC}_4$ has an ER-labeling where the labels on the set $\mathcal{M}_{\mathcal{NC}_4}$ of maximal chains of $\mathcal{NC}_4$ correspond to parking functions of length $3$, see Figure~\ref{fig:NC4}. So $\mathcal{NC}_4$
has $16$ maximal chains with label words given by $(1,1,1)$, the three permutations of each $(1,1,2)$ $(1,1,3)$ and $(1,2,2)$; and the six permutations of $(1,2,3)$.  Considering the descent sets of each of these sequences we can compute that $\beta_{\mathcal{NC}_4}(\emptyset) = 1$,  $\beta_{\mathcal{NC}_4}(\{1\}) = 5$,  $\beta_{\mathcal{NC}_4}(\{2\}) = 5$ and $\beta_{\mathcal{NC}_4}(\{1,2\}) = 5$.  Thus,
$$
F_{\mathcal{NC}_4}(\mathbf{x}) = L_{\emptyset, 3}(\mathbf{x})+5L_{\{1\},3}(\mathbf{x})+5L_{\{2\},3}(\mathbf{x})+5L_{\{1,2\},3}(\mathbf{x}).
$$
The quasisymmetric function $F_{\mathcal{NC}_n}(\mathbf{x})$ is in fact symmetric. Stanley~\cite{Stanley1997} showed that $\omega(F_{\mathcal{NC}_n}(\mathbf{x}))$ is \emph{Haiman's Parking Function Symmetric Function} of $n$, where $\omega$ is the involution on the
ring of quasisymmetric functions given by  $\omega(L_{S,n}) = L_{S^c, n}$ where $S^c$ is the
complement
of $S$ in $[n-1]$.

\end{example}

\begin{example}
Now consider $F_P(\mathbf{x})$ when $P=\mathcal{NCD}yck_4\cong Q_{\lambda}(\mathcal{NC}_4)$ together with its inherited  ER$^*$-labeling from $\mathcal{NC}_4$ in Section \ref{section:noncrossingpartitiondual}. Using Proposition  \ref{proposition:bijectionbetweensaturatedchains} we know that the maximal chains are in a label-preserving bijective correspondence with the ones of $\mathcal{NC}_4$, so they are labeled by parking functions as well.  One can show then that $F_{\mathcal{NCD}yck_4}(\mathbf{x}) = 5L_{\emptyset, 3}(\mathbf{x}) + 5L_{\{1\},3}(\mathbf{x})+5L_{\{2\},3}(\mathbf{x})+ L_{\{1,2\},3}(\mathbf{x})$.
\end{example}

The reader may have noticed that the quasisymmetric functions above are very closely
related.   Our examples show that
$$
F_{\Pi_3}(\mathbf{x}) = \omega (F_{\mathcal{ISF}_3}(\mathbf{x})) \text{ and } F_{\mathcal{NC}_4}(\mathbf{x}) = \omega(F_{\mathcal{NCD}yck_4}(\mathbf{x})).
$$

This is no coincidence as we now see.
\begin{theorem}\label{FandOmegaF}
Let $\lambda$ be a generalized CW-labeling of $P$. Then
$$ F_{Q_{\lambda}(P)}(\mathbf{x})=\omega(F_P(\mathbf{x})).$$
\end{theorem}

\begin{proof}
Recall that since $\lambda$ is a CR-labeling, $\beta_P(S)$ counts the number
of maximal chains with descent set $S$.  Similarly since $\lambda^*$ is an ER$^*$-labeling of
$Q_{\lambda}(P)$, $\beta_{Q_{\lambda}(P)}(S)$ is the number of maximal chains with ascent set
$A(\c)
:= \{i \mid \lambda(\c, x_{i-1},x_i) < \lambda(\c, x_i,x_{i+1})\}=S$.  The
CW-analogue of Proposition~\ref{proposition:bijectionbetweensaturatedchains} implies that there is
a bijection between maximal chains in $P$ and $Q_{\lambda}(P)$ which preserves labels.  It follows
that for each $S\subseteq [n-1]$,
$$
\sum_{m\in P} \beta_{[\hat{0},m]}(S^c)= \sum_{m' \in Q_{\lambda}(P)}\beta_{[\hat{0},m']}(S)
$$
where each sum is over maximal elements of $P$ and $Q_{\lambda}(P)$ respectively.
Therefore
\begin{align*}
 \omega F_{Q_{\lambda}(P)}(\mathbf{x}) &=  \omega\left( \sum_{m'\in Q_{\lambda}(P)} F_{[\hat{0},m']}(\mathbf{x})
\right)\\
  &=  \omega\left(  \sum_{m'\in Q_{\lambda}(P)} \left(\sum_{S\subseteq [n-1]}
\beta_{[\hat{0},m']}(S) L_{S,n}\right) \right)\\
  &=  \omega  \left(\sum_{S\subseteq [n-1]} \left(  \sum_{m'\in
Q_{\lambda}(P)}\beta_{[\hat{0},m']}(S)\right) L_{S,n} \right)\\
&=  \omega\left( \sum_{S\subseteq [n-1]}\left(  \sum_{m\in P}\beta_{[\hat{0},m]}(S^c)\right)
L_{S,n}\right)\\
&=  \sum_{S\subseteq  [n-1]} \left(  \sum_{m\in P}\beta_{[\hat{0},m]}(S^c) \right) \omega(L_{S,n})\\
&=\sum_{S\subseteq [n-1]} \left(  \sum_{m\in P}\beta_{[\hat{0},m]}(S^c) \right) L_{S^c,n}\\
&=\sum_{m\in P} \left(\sum_{S\subseteq [n-1]} \beta_{[\hat{0},m]}(S) L_{S,n}\right)\\
&= F_P(\mathbf{x})
\end{align*}
Thus we have proved the desired result.
\end{proof}

\subsection{The characteristic of the action}

In~\cite{Norton1979}, Norton investigated  the representation theory of $H_n(0)$.  It is known that
there are $2^{n-1}$ irreducible representations, all of them one-dimensional and hence they can be
indexed by subsets of $[n-1]$.  With this indexing,  we have that if $U_i$ is one of the generators
of $H_n(0)$ then the representation $\psi_S$ is given by
$$
\psi_S(U_i)=
\begin{cases}
1& \mbox{ if $i\in S,$}\\
0 & \mbox{otherwise.}
\end{cases}
$$
Hence, the character of the action is given by
$$
\chi_S(U_{i_1}U_{i_2}\cdots U_{i_k})=
\begin{cases}
1& \mbox{ if $i_1, i_2,\dots, i_k\in S,$}\\
0 & \mbox{otherwise.}
\end{cases}
$$
The \emph{(quasisymmetric) characteristic} of the character $\chi_S$ is defined by
$$
ch(\chi_S) = L_{S,n}
$$
where, as before, $L_{S,n}$ is Gessel's fundamental quasisymmetric function.  We will
use $\chi_P$ to denote the character of the defining representation of a $H_n(0)$-action on $P$.

\begin{theorem}\label{CharacteristicOfAction}
Let $P$ be a graded poset of rank $n$ with a generalized CW-labeling.  The local $H_n(0)$-action
previously described is such that
$$
ch(\chi_P)=F_P(\mathbf{x}).
$$
\end{theorem}
We note that the proof we present is almost identical to the one in~\cite[Proposition 4.1]{McNamara2003}.
\begin{proof}
Let $[L_{S,n}] f(\mathbf{x})$ denote the coefficient of $L_{S,n}$ in the expansion of the quasisymmetric
function $f(\mathbf{x})$ in the fundamental basis.  We will show for any subset $S$
of $[n-1]$,
$$
 [L_{S,n}]ch(\chi_P)=[L_{S,n}]F_p(\mathbf{x}).
$$
 As we saw earlier, the coefficient in $F_P(\mathbf{x})$ is
$$
\sum_{m} \beta_{[\hat{0}, m]}(S).
$$
    Thus, it suffices to show
$$
 [L_{S,n}]ch(\chi_P)=\sum_{m} \beta_{[\hat{0}, m]}(S).
$$

Now let $J\subseteq [n-1]$ and let $\{i_1,i_2,\dots, i_k\}$ be a multiset of $J$ where each element
of $J$ appears at least once.  For $\c\in \mathcal{M}_P$, if $U_i(\c) \neq
\c$, then $\c$ has an ascent at $i$.   It follows that $U_{i_1}U_{i_2}\cdots
U_{i_k}(\c)=\c$ if and only if $\c$ has descent set containing $J$.
Therefore,
\begin{align*}
\chi_P(U_{i_1}U_{i_2}\cdots U_{i_k})&= \#\{\c\in \mathcal{M}_P \mid D(\c)\supseteq J\}\\
&= \sum_{S\supseteq J} \#\{\c\in \mathcal{M}_P \mid D(\c)=S\}\\\
&= \sum_{m} \sum_{S\supseteq J} \#\{\c\in \mathcal{M}_{[\hat{0},m]} \mid D(\c)=S\}\\\
&= \sum_{m} \left(\sum_{S\supseteq J} \beta_{[\hat{0},m]}(S) \right)\\
&= \sum_{S\subseteq [n-1]} \left(\sum_{m} \beta_{[\hat{0},m]}(S) \right)\chi_S(U_{i_1}U_{i_2}\cdots
U_{i_k})
\end{align*}

It follows that
$$
[L_{S,n}] ch(\chi_P) = [L_{S,n}]  ch\left(  \sum_{S\subseteq [n-1]} \left(\sum_{m}
\beta_{[\hat{0},m]}(S) \right)\chi_S\right) = \sum_{m} \beta_{[\hat{0},m]}(S)
$$
which completes the proof.
\end{proof}

The analogue of Proposition~\ref{proposition:bijectionbetweensaturatedchains} for CW-labelings
implies that there is a bijection between maximal chains of $P$ and $Q_{\lambda(P)}$  which
preserves labels.  It follows that the local $H_n(0)$-action on
$\mathbb{C}\mathcal{M}(P)$ can be also transported to a $H_n(0)$-action on
$\mathbb{C}\mathcal{M}(Q_{\lambda})$.  It turns out that this action on the maximal chains of $Q_\lambda(P)$ is local.

\begin{lemma}
Let $P$ be a graded poset with a generalized CW-labeling $\lambda$.  The $0$-Hecke algebra action
on $Q_\lambda(P)$ is local.
\end{lemma}
\begin{proof}
We must show that if we apply $U_i$  to any maximal chain $\d$ of $Q_\lambda(P)$, the chain we get
agrees with $\d$ everywhere except possible at rank $i$.   Suppose that $\d$ and  $\d'
$ are maximal chains in $Q_\lambda(P)$ such that $U_i(\d) = \d'$ and $\d\neq \d'$.  Let $\c$ and
$\c'$ be respectively the preimages of these chains under the label preserving bijection between
$\mathcal{M}_{Q_\lambda(P)}$ and $\mathcal{M}_P$ described in the generalized CW-labeling version of
Proposition~\ref{proposition:bijectionbetweensaturatedchains}.  Then $U_i(\c) =\c'$ and $\c \neq
\c'$  Since the action on $P$ is local, we can write $\c: (\hat{0}=x_0\lessdot x_1\lessdot \cdots
x_{i-1}\lessdot x_i \lessdot x_{i+1} \lessdot \cdots \lessdot x_n)$ and $\c': (\hat{0}=x_0\lessdot x_1\lessdot
\cdots x_{i-1}\lessdot x'_i \lessdot x_{i+1} \lessdot \cdots \lessdot x_n)$.  Denote $\c_k$ the
subchain formed by the smallest $k+1$ elements of $\c$ and $\c'_k$ the one formed by the smallest
$k+1$ elements of $\c'$.  We have that $\c_j = \c'_j$ for all $0\leq j\leq i-1$ and $\c_i \neq
\c'_i$, but $\c_j $ and $\c'_j$ are equivalent for $j\geq i+1$ (since one chain is obtained from
the other after applying a quadratic exchange at rank $i$).  Thus the chains
$[\c_0]\lessdot [\c_1]\lessdot \cdots \lessdot [\c]$ and $[\c'_0]\lessdot [\c'_1]\lessdot \cdots \lessdot [\c']$
 in $Q_\lambda(P)$ agree everywhere except at rank $i$.  Moreover, by the (generalized
CW-labeling versions of) Lemma \ref{lemma:main} and the proof of
Proposition~\ref{proposition:bijectionbetweensaturatedchains} these chains are exactly  $\d$ and
$\d'$.  We conclude that the action on $Q_\lambda(P)$ is local.
\end{proof}

\begin{proposition}\label{charOfQ}
Let $P$ be a poset with a generalized CW-labeling $\lambda$.  For any maximal interval $I$ in
$Q_\lambda(P)$,
$$
ch(\chi_I) = \omega(F_I(\mathbf{x}))
$$
\end{proposition}
\begin{proof}
First note that since the action on $Q_\lambda(P)$ is local,  the action only permutes maximal
chains within maximal intervals of $Q_\lambda(P)$.   Also, just as with $P$,  $U_i(\c) \neq \c'$ if
and only if $\c$ has ascent at rank level $i$.    Finally,  note that since the labeling on
$Q_\lambda(P)$ (and hence on $I$) is an ER$^*$-labeling, we have that $\beta_{I}(S) =\{\c \in
\mathcal{M}_I\mid D(\c) =S^c\}$ for any $S\subseteq [n-1]$.   With this in mind, one can check that
a slight modification of the proof of Theorem~\ref{CharacteristicOfAction} gives the result.
\end{proof}

\begin{remark}
Note that $F_{Q_{\lambda}(P)}(\mathbf{x})=\sum_{I}(F_I(\mathbf{x}))$ and that $\chi_{P}=\chi_{Q_{\lambda}(P)}=\sum_{I} \chi_I$, where the sums are over maximal intervals $I$ of $Q_{\lambda}(P)$. Hence we obtain  Theorem
\ref{theorem:goodactions} as a corollary of Theorems \ref{FandOmegaF}  and \ref{CharacteristicOfAction}; and Proposition \ref{charOfQ}.
\end{remark}

A poset $P$  is called \emph{bowtie-free} if  there does not exist distinct $a,b,c,d\in P$ with $c
\lessdot a$, $d \lessdot a$, $c \lessdot b$ and $d \lessdot b$.
In~\cite{McNamara2003}, McNamara showed that a bowtie-free poset $P$ with a $\hat{0}$ and a
$\hat{1}$ has a local
$H_n(0)$-action with the property that the characteristic of this action is $\omega(F_p(\mathbf{x}))$ if and
only if $P$ is snellable.   Additionally, he showed that if $P$ is a lattice, then $P$ is
supersolvable.  Proposition~\ref{charOfQ} then   implies the following corollary.

\begin{corollary}\label{corollary:mcnamara}
Let $P$ be a poset with a generalized CW-labeling $\lambda$.  If $I$ is a maximal interval of
$Q_\lambda(P)$
and is bowtie free, then $I$ is snellable.  Moreover, if $I$ is a lattice, then $I$ is
supersolvable.
\end{corollary}

\section{Open questions and further work}
In this Section we leave a few open questions that are motivated by the present work.
In Theorem \ref{theorem:WimpliesWhitney} we showed that posets that have Whitney labelings
also have Whitney duals. It is reasonably to expect that there are posets without Whitney labelings
that have Whitney duals. Indeed, the poset $\mathcal{ISF}_3$ has $\Pi_3$ as a Whitney dual.
However $\mathcal{ISF}_3$ cannot have a Whitney labeling since in one of the maximal intervals the rank two
switching property cannot be satisfied, see Figure \ref{figure:examplemobius}. We would like to
know if there is a general characterization of graded posets that have Whitney duals that
completely answers Question \ref{question:whitneydual1}.  Additionally, we would like to know if there are other
different and insightful methods of constructing Whitney duals, we propose the following question.
\begin{question}
 Is there a systematic way to construct a Whitney dual of $P$ without the use of labelings?
\end{question}

In the context of Whitney labelings we have provided two definitions: Whitney labelings and
generalized Whitney labelings. Although the conditions of a generalized Whitney labeling are the
ones we use in the proofs, all our examples satisfy the, a priori stronger, requirements of Whitney
labelings.

\begin{question}
Are the families of Whitney labelable graded posets and of generalized Whitney labelable
graded posets equal?
\end{question}

Of main interest is also to better understand the structure of the posets $Q_{\lambda}(P)$ that are
constructed using Whitney labelings $\lambda$ of $P$. We know from Theorem
\ref{theorem:Qlambdanonisomorphic} that these posets are strongly dependent on $\lambda$ and in
Theorem \ref{theorem:secondcaracterizationQlambda} we provide a different description of its
poset structure.
\begin{question}
 Is there a nice way of characterizing all the posets that are of the form $Q_{\lambda}(P)$ for some
poset $P$ and some Whitney labeling $\lambda$?
\end{question}

In light of Corollary \ref{corollary:mcnamara} determining the structural properties of the posets $Q_{\lambda}(P)$ also becomes relevant.

\begin{question}
Are all $Q_{\lambda}(P)$ lattices? If this is not the case, are all of them bowtie-free?
\end{question}

\section*{Acknowledgments}
The authors are extremely grateful to Michelle Wachs for the various stimulating discussions that led to the concept of Whitney duality and this project. The authors are also very thankful to Peter McNamara for many useful conversations.

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