Ehrhart $h^{\ast }$-vectors of hypersimplices

Abstract

We consider the Ehrhart $h^{\ast }$-vector for the hypersimplex. It is well-known that the sum of the $h_i^{\ast }$ is the normalized volume which equals an Eulerian numbers. The main result is a proof of a conjecture by R. Stanley which gives an interpretation of the $h_i^{\ast }$ coefficients in terms of descents and excedances. Our proof is geometric using a careful book-keeping of a shelling of a unimodular triangulation. We generalize this result to other closely related polytopes.

References in corpus (1)

• The Hilbert series of algebras of Veronese type

Cited by in corpus (1)

• An Eulerian permutation statistic and generalizations