 
Summary: J. reine angew. Math. 583 (2005), 163174 Journal fu¨r die reine und
angewandte Mathematik
( Walter de Gruyter
Berlin Á New York 2005
Ehrhart polynomials, simplicial polytopes, magic
squares and a conjecture of Stanley
Dedicated to Richard Stanley on the occasion of his sixtieth birthday
By Christos A. Athanasiadis at Heraklion
Abstract. It is proved that for a certain class of integer polytopes P the polynomial
hðtÞ which appears as the numerator in the Ehrhart series of P, when written as a rational
function of t, is equal to the hpolynomial of a simplicial polytope and hence that its co
ecients satisfy the conditions of the gtheorem. This class includes the order polytopes
of graded posets, previously studied by Reiner and Welker, and the Birkho¤ polytope of
doubly stochastic n Â n matrices. In the latter case the unimodality of the coecients of
hðtÞ, which follows, was conjectured by Stanley in 1983.
1. Introduction
Let P be an mdimensional convex polytope in Rq
having integer vertices. We will be
concerned with the function iðP; rÞ counting integer points in the rfold dilate of P. It is a
fundamental result due to Ehrhart [3], [4] that iðP; rÞ is a polynomial in r of degree m, called
