Interpreting the Ehrhart coefficients of cross-polytopes

TL;DR

This paper provides a combinatorial proof that Ehrhart polynomials of cross-polytopes have positive coefficients by linking them to generating functions for colored permutations, addressing a question by Stanley.

Contribution

It introduces a combinatorial interpretation of Ehrhart coefficients of cross-polytopes using colored permutations, confirming their positivity.

Findings

Ehrhart polynomials of cross-polytopes have positive coefficients

A scaled version of these polynomials are generating functions for colored permutations

Answers a question posed by Stanley regarding Ehrhart coefficients

Abstract

It is known that the Ehrhart polynomials of cross-polytopes, as well as of pyramids over them, have positive coefficients. We give a combinatorial proof of this fact by showing that a scaled version of the Ehrhart polynomials are generating functions for certain colored permutations. This answers a question posed by Stanley.