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Summary: J. reine angew. Math. 583 (2005), 163--174 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 h-polynomial of a simplicial polytope and hence that its co-
ecients satisfy the conditions of the g-theorem. This class includes the order polytopes
of graded posets, previously studied by Reiner and Welker, and the Birkho¤ polytope of
doubly stochastic n  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 m-dimensional convex polytope in Rq
having integer vertices. We will be
concerned with the function iðP; rÞ counting integer points in the r-fold 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
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