H-polynomial of the half-open hypersimplex

Details H-polynomial of the half-open hypersimplex H-polynomial of the half-open hypersimplex

2011-04-28

arxiv.org/abs/1104.5292v1

Mathematics

Combinatorics

Scientific paper

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.

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