Ehrhart h[superscript ∗]-Vectors of Hypersimplices

Abstract

We consider the Ehrhart h[superscript ∗]-vector for the hypersimplex. It is well-known that the sum of the h[superscript ∗][subscript i] is the normalized volume which equals the Eulerian numbers. The main result is a proof of a conjecture by R. Stanley which gives an interpretation of the h[superscript ∗][subscript i] coefficients in terms of descents and exceedances. 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.