Ehrhart $h^*$-vectors of hypersimplices

TL;DR

This paper proves a conjecture relating Ehrhart h*-vector coefficients of hypersimplices to descents and excedances, using geometric shelling techniques, and extends the result to related polytopes.

Contribution

It provides a geometric proof of Stanley's conjecture on Ehrhart h*-vectors and generalizes the interpretation to other polytopes.

Findings

Confirmed the combinatorial interpretation of h*-vector coefficients.

Developed a shelling-based geometric proof technique.

Extended the interpretation to related polytopes.

Abstract

We consider the Ehrhart $h^*$-vector for the hypersimplex. It is well-known that the sum of the $h_i^*$ 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 $h^*_i$ 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.