Ehrhart polynomials, simplicial polytopes, magic squares and a conjecture of Stanley

Abstract

It is proved that for a certain class of integer polytopes P the polynomial h(t ) which appears as the numerator in the Ehrhart series of P, when written as a rational function of t, is equal to the h-polynomial of a simplicial polytope and hence that its co-efficients satisfy the conditions of the g-theorem. This class includes the order polytopes of graded posets, previously studied by Reiner and Welker, and the Birkhoff polytope of doubly stochastic n × n matrices. In the latter case the unimodality of the coefficients of h (t ), which follows, was conjectured by Stanley in 1983.