$h^\ast$-polynomials of zonotopes

Abstract

The Ehrhart polynomial of a lattice polytope $P$ encodes information about the number of integer lattice points in positive integral dilates of $P$. The $h^\ast$-polynomial of $P$ is the numerator polynomial of the generating function of its Ehrhart polynomial. A zonotope is any projection of a higher dimensional cube. We give a combinatorial description of the $h^\ast$-polynomial of a lattice zonotope in terms of refined descent statistics of permutations and prove that the $h^\ast$-polynomial of every lattice zonotope has only real roots and therefore unimodal coefficients. Furthermore, we present a closed formula for the $h^\ast$-polynomial of a zonotope in matroidal terms which is analogous to a result by Stanley (1991) on the Ehrhart polynomial. Our results hold not only for $h^\ast$-polynomials but carry over to general combinatorial positive valuations. Moreover, we give a complete description of the convex hull of all $h^\ast$-polynomials of zonotopes in a given dimension: it is a simplicial cone spanned by refined Eulerian polynomials.