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

TL;DR

This paper proves that a symmetric sequence related to magic squares and Ehrhart polynomials corresponds to the h-vector of a simplicial polytope, confirming Stanley's unimodality conjecture and extending the result.

Contribution

It establishes a connection between magic square enumeration, Ehrhart polynomials, and the g-theorem by identifying the sequence as an h-vector of a simplicial polytope.

Findings

The sequence satisfies the conditions of the g-theorem.

The sequence is unimodal, confirming Stanley's conjecture.

Several generalizations of the main result are provided.

Abstract

It is proved that a certain symmetric sequence 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 by 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 this sequance, which follows, was conjectured by Stanley (1983). Several generalizations are given.