Ehrhart Polynomials, Simplicial Polytopes, Magic Squares and a Conjecture of Stanley

It is proved that a certain symmetric sequence (h0, h1, . . . , hd) of nonnegative integers arising in the enumeration of magic squares of given size n by row sums or, equivalently, in the generating function of the Ehrhart polynomial of the polytope of doubly stochastic n × n matrices, is equal to the h-vector of a simplicial polytope and hence that it satisfies the conditions of the g-theorem. The unimodality of (h0, h1, . . . , hd), which follows, was conjectured by Stanley (1983). Several generalizations are given.