## Posets of finite sequences are highly connected

### Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

I need the following result for an example in a paper I'm writing. It's easy enough to prove, but I'd prefer to just give a reference. Does anyone know one?

Fix $1 \leq k \leq n$. Define $X_{n,k}$ to be the following poset. The elements of $X_{n,k}$ are ordered sequences $\omega = (x_1,\ldots,x_m)$, where the $x_i$ are distinct elements of the $n$-element set `$\{1,\ldots,n\}$` and $m \geq k$. The order relation is that $\omega_1 \leq \omega_2$ if $\omega_1$ is a subsequence of $\omega_2$. The theorem then is that the geometric realization of $X_{n,k}$ is $(n-1-k)$-connected.

-
what's the geometric realization of a poset ? – Suresh Venkat Oct 13 2010 at 20:47

1
The set of chains is an abstract simplicial complex, so the geometric realization is just the realization of this complex. – Richard Kent Oct 13 2010 at 20:54
ah ok. i was wondering if that is what it was. – Suresh Venkat Oct 13 2010 at 21:44

## 1 Answer

If I understand your question correctly, an answer should appear in the paper "On lexicographically shellable posets" of Anders Bj\"orner and Michelle Wachs, in Transactions of the AMS 277, pp. 323-341.

-

Yes, this paper gives me exactly what I want.  Thanks! – Andy Putman Oct 14 2010 at 1:39
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 12, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9114333391189575, "perplexity_flag": "head"}