# Tagged Questions

Questions related to the different ways of expressing an integer as a sum of integers; or, questions related to the subdivision of a set into smaller disjoint sets; questions related to the subdivision of an interval into smaller intervals that intersect only at the endpoints.

1answer
70 views

### HAKMEM 18(B): Cubic Partitions

Taken from HAKMEM 18. Quoting... A partition of $N$ is a finite string of non-increasing integers that add up to $N$. Thus 7 3 3 2 1 1 1 is a partition of 18. Sometimes an infinite string of ...
2answers
248 views

### A question on partitions of n

Let $P$ be the set of partitions of n. Let $\lambda$ denote the shape of a particular partition. Let $f_\lambda(i)$ be the frequency of $i$ in $\lambda$ and let \$a_\lambda(i) := \# \lbrace j : ...
3answers
396 views

### An Inequality Involving Bell Numbers:  $B_n^2 \leq B_{n-1}B_{n+1}$

The following inequality came up while trying to resolve a conjecture about a certain class of partitions (the context is not particularly enlightening): $$B_n^2 \leq B_{n-1}B_{n+1}$$ for \$n \geq ...
2answers
292 views

### How can I reduce a number?

I'm trying to work on a program and I think I've hit a math problem (if it's not, please let me know, sorry). Basically what I'm doing is taking a number and using a universe of numbers and I'm ...
1answer
296 views

### Number of solutions of  Frobenius equation

I have one problem which needs to count the number of solution of the equation $$2x+7y+11z=42$$ where $x,y,z \in \{0,1,2,3,4,5,\dots\}$. My attempt: I noticed that that maximum value of $z$ could ...
1answer
261 views

### Recurrence for the partition numbers

I'm reading Analytic Combinatorics [PDF] book by Flajolet and Sedgewick, and I can't figure out one of the steps in the derivation of the $P_n$ — number of partitions of size $n$ (or coefficients in ...
2answers
200 views

### Two combinatorics problems. I'm not 100% confident in my answers

These are two problems from my combinatorics assignment that I'm not quite confident in my answer. Am I thinking of these the right way? Problem 1: On rolling 16 dice. How many of the $6^{16}$ ...
2answers
298 views

### Learning about partitions and modular forms

I'm interested in learning about partitions and modular forms. I already know algebra and analysis (complex and real). Can any one suggest me books or other materials from where I can learn these ...
0answers
121 views

### Jordan Measure, Semi-closed sets and Partitions

From earlier question here. Consider $$\mu \left( (0,1) \cap \mathbb Q \right) = 0$$ where $$(0,1) = (0,\frac{1}{3})\cup [\frac{1}{3}, \frac{2}{3}] \cup (\frac{2}{3}, 1)$$ so  \mu ((0,1) \cap ...
2answers
248 views

### Jordan Measures, Open sets, Closed sets and Semi-closed sets

I cannot understand: $$\bar{\mu} ( \{Q \cap (0,1) \} ) = 1$$ and (cannot understand this one particularly) $$\underline{\mu} ( \{Q \cap (0,1) \} ) = 0$$ where $Q$ is rational numbers, why? I ...
1answer
744 views

### Ellipse 3-partition: same area and perimeter

Inspired by the question, "How to partition area of an ellipse into odd number of regions?," I ask for a partition an ellipse into three convex pieces, each of which has the same area and the same ...
1answer
971 views

### The number of partitions of $n$ into distinct parts equals the number of partitions of $n$ into odd parts

This seems to be a common result. I've been trying to follow the bijective proof of it, which can be found easily online, but the explanations go over my head. It would be wonderful if you could give ...
1answer
369 views

### Partition number problem

Denote by $I_m=\{0,1,2,…m\}$, by $N_s=\{1,2,…,s\}$ , by $\overline s$ least common multiple of elements of set $N_s$ and by $p(k,N_s)$ the number of partitions of natural number $k$ in parts used ...
1answer
152 views

### Number of ways to sum square numbers to yield a given number

I would like to know how many choices of $x_i$ there are such that $$\sum_{i=1}^{n}x_i^2=m$$ where $n$, $m$ are given. The $x_i$ can be any nonnegative integer and need not be unique and the order is ...
2answers
438 views

### Partition Problem, verifying solution in polynomial time

I add a look at the partition problem, this problem is know as the Easiest hard problem since it is NP-complete and seems pretty easy. From wikipedia on NP-complete: In computational complexity ...
1answer
712 views

### Hardy Ramanujan Asymptotic Formula for the Partition Number

I am needing to use the asymptotic formula for the partition number, $p(n)$ (see here for details about partitions). The asymptotic formula always seems to be written as, \$ p(n) \sim ...
1answer
93 views

### Something basic in “l-adic properties of the partition function” paper

I am trying to understand the basic result in this paper: http://www.aimath.org/news/partition/folsom-kent-ono.pdf My problem is with the example at the end of page 2. I understand it's supposed to ...
1answer
198 views

### Closed-form Expression of the Partition Function $p(n)$

I feel like I have seen news that a paper was recently published, at most a few months ago, that solved the well-known problem of finding a closed-form expression for the partition function $p(n)$ ...
2answers
81 views

### List all 3 part compositions of 5

I am looking at a past exam written by a student. There was a question I believed he got correct but received only 1/4. The marker wrote down "4 more compositions, order matters". This is the ...
2answers
150 views

### Trouble understanding proof that the unit interval cannot be partitioned in a certain way

From the book "Putnam and Beyond." The problem: Show that the interval [0, 1] cannot be partitioned into two disjoint sets A and B such that B = A + a for some real number a. Proof: Assume ...
0answers
45 views

### Prove that single machine can reduce from 2-partition problem

I'm given the following problem: Prove that single machine \sum U_i (number of tardy job) with release date constraints problem can reduce from 2-partition problem ...
2answers
97 views

### Generating sequences of numeric partitions

Definition: A tuple $\lambda = (\lambda_1, \ldots, \lambda_k)$ of natural numbers is called a numeric partition of $n$ if $1 \leq \lambda_1 \leq \cdots \leq \lambda_k$ and \$\lambda_1 + \cdots + ...
1answer
293 views

### identity proof for partitions of natural numbers

Definition: A tuple $\lambda = (\lambda_1, \cdots, \lambda_k)$ of Natural Numbers is called a numeric partition of n if $1 \leq \lambda_1 \leq \cdots \leq \lambda_k$ and \$\lambda_1 + \cdots + ...
1answer
295 views

### How to find the coefficient of a term in this expression

How to determine the coefficient of z3q100 in I stumbled upon this problem while trying to solve this type of partition problem: Find the number of integer solutions to x + y + z = 100 such that 3 ...
1answer
326 views

### Number of permutations with a given partition of cycle sizes

Part of my overly complicated attempt at the Google CodeJam GoroSort problem involved computing the number of permutations with a given partition of cycle sizes. Or equivalently, the probability of a ...
1answer
153 views

### Why is there a derivative in this formula?

This is a very simple question. Why is Rademacher's formula presented with d/dx in it? Why not just "do" the derivative? Then replace x with n? Is it so there is only one transcendental ...
1answer
427 views

### Feeding real or even complex numbers to the integer partition function $p(n)$?

Like most people, when I first encountered $n!$ in grade school, I graphed it, then connected the dots with a smooth curve and reasoned that there must be some meaning to $\left(\frac43\right)!$ — ...
1answer
131 views

### What is the standard way of writing this partition function? [closed]

What is the standard way of writing the number of partitions of n where each number is less than or equal to k. Is it $p(n,k)$ or $p(k,n)$? So far in my reading, have seen both.
3answers
261 views

### Partition an integer $n$ by limitation on size of the partition

According to my previous question, is there any idea about how I can count those decompositions with exactly $i$ members? for example there are $\lfloor \frac{n}{2} \rfloor$ for decompositions of $n$ ...
3answers
166 views

### Decomposition by subtraction

In how many ways one can decompose an integer $n$ to smaller integers at least 3? for example 13 has the following decompositions: \begin{gather*} 13\\ 3,10\\ 4,9\\ 5,8\\ 6,7\\ 3,3,7\\ 3,4,6\\ ...
1answer
143 views

### Seeking some details about what is denoted by the partition function $P(n,k)$

Quoting from Wolfram MathWorld, "$P(n,k)$ denotes the number of ways of writing $n$ as a sum of exactly $k$ terms or, equivalently, the number of partitions into parts of which the largest is exactly ...
2answers
180 views

### Asymptotics for partitions of $n$ with largest part at most $k$ (or into at most $k$ parts)

Let $\bar p_k(n)$ be the number of partitions of $n$ with largest part at most $k$ (equivalently, into at most $k$ parts). Is there an elementary formula for the asymptotic behavior of $\bar p_k(n)$ ...
1answer
175 views

### Seeking a textbook proof of a formula for the number of set partitions whose parts induce a given integer partition

Let $t \geq 1$ and $\pi$ be an integer partition of $t$. Then the number of set partitions $Q$ of $\{1,2,\ldots,t\}$ for which the multiset $\{|q|:q \in Q\}=\pi$ is given by \[\frac{t!}{\prod_{i \geq ...
1answer
88 views

### Validity of a q-series theorem

Define the $q$-analog $(a;q)_n = \prod_{k=0}^n \left(1 - aq^k\right)$. I want to prove the identity $\frac{(q^2;q^2)_\infty}{(q;q)_\infty}=\frac{1}{(q;q^2)_\infty}$. I viewed the LHS this way: ...
2answers
263 views

### Natural set to express any natural number as sum of two in the set

Any natural number can be expressed as the sum of three triangular numbers, or as four square numbers. The natural analog for expressing numbers as the sum of two others would apparently be the sum ...
1answer
179 views

### What is the number of ways to choose x groups from y items? (partitions with x cells of a multiset)

Where a group can consist of 1 or more items, groups don't have to be equally sized and items can be duplicates. Example - Choose 3 groups: Items: 1 2 2 3 Groups: (1) (2 2) (3) (1 2) (2) (3) (3 ...
2answers
233 views

### Graph coloring problem (possibly related to partitions)

Given an undirected graph I'd like to color each node either black or red such that at most half of every node's neighbors have the same color as the node itself. As a first step, I'd like to show ...
1answer
436 views

### Upper bound on integer partitions of n into k parts

Recent news piqued my interest in integer partitions again. I'm working my way back through an old text and I'm completely hung up on this problem: Recall that $p_k(n)$ is the number of partitions ...
1answer
46 views

### Notation for “duplicating” partitions

I'm using Macdonald's "Symmetric Functions and Hall Polynomials" as a reference and did not find what I was looking for -- apologies if I only missed it. As an example, let us consider the partition ...
1answer
772 views

### On problems of coins totaling to a given amount

I don't know the proper terms to type into Google, so please pardon me for asking here first. While jingling around a few coins, I realized that one nice puzzle might be to figure out which $n$ or so ...
0answers
173 views

### Visualizing the Partition numbers (suggestions for visualization techniques)

So Ken Ono says that the partition numbers behave like fractals, in which case I'd like to try to find an appropriately illuminating way of visualizing them. But I'm sort of stuck at the moment, so ...
4answers
2k views

### Algorithm for generating integer partitions

I'm looking for a fast algorithm for generating all the partitions of an integer up to a certain maximum length; ideally, I don't want to have to generate all of them and then discard the ones that ...
6answers
2k views

### Making Change for a Dollar (and other number partitioning problems)

I was trying to solve a problem similar to the "how many ways are there to make change for a dollar" problem. I ran across a site that said I could use a generating function similar to the one quoted ...
1answer
108 views

### Number of distributions leaving none of $n$ cells empty

The solution for the number of distributions leaving none of the $n$ cells empty (with unlike cells and $r$ unlike objects) is given by ...
3answers
284 views

### “Converting” equivalence relations to partitions

There is a direct relationship between equivalence relations and partitions. Is there a way to simply use an equivalence relation's definition to get the matching partition? And what about the other ...
1answer
323 views

### Matrix representation of a partition

Is there a natural way to represent all the partitions of an integer set {1,2,3,...,n} as a matrix in the similar way permutations can be mapped to group of matrices?
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