math.CO

(what is this?)

# Title:H-polynomial of the half-open hypersimplex

Authors:Nan Li
Abstract: We study the h-polynomial of the half-open hypersimplex $\Delta'_{k,n}$, defined as the slice of the hypercube $[0,1]^{n-1}$ such that $k-1<\sum_{i=1}^{n-1}x_i\le k$. It is well-known that the sum of all the coefficients in the h-polynomial is the normalized volume of $\Delta'_{k,n}$, which is the Eulerian number $A_{k,n-1}$. The main result of this paper was conjectured by Stanley and says that the coefficient of $t^s$ in the h-polynomial is the number of permutations with $n-1$ letters, $k-1$ excedances and $s$ descents. We have two proofs: first by a generating function method, based on a formula by Foata and Han for the joint distribution of descents and excedances, and second by a shellable triangulation.
 Subjects: Combinatorics (math.CO) MSC classes: 05A05, 52B22, 52C07, 05A15, 06F99, 05C38 Cite as: arXiv:1104.5292 [math.CO] (or arXiv:1104.5292v1 [math.CO] for this version)

## Submission history

From: Nan Li [view email]
[v1] Thu, 28 Apr 2011 04:09:50 UTC (20 KB)
[v2] Thu, 9 Aug 2012 09:06:29 UTC (29 KB)
{"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": 1, "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, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5697028636932373, "perplexity": 1159.102794192308}, "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}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232258620.81/warc/CC-MAIN-20190526004917-20190526030917-00009.warc.gz"}