Hypersimplices are Ehrhart Positive

TL;DR

This paper proves that hypersimplices have Ehrhart polynomials with positive coefficients, providing a combinatorial formula and confirming Ehrhart positivity for uniform matroids, thus resolving a longstanding conjecture.

Contribution

It establishes Ehrhart positivity for hypersimplices and uniform matroids, introducing weighted Lah numbers and their properties as key tools.

Findings

Ehrhart polynomials of hypersimplices have positive coefficients.

A combinatorial formula for these coefficients is provided.

Uniform matroids are shown to be Ehrhart positive.

Abstract

We consider the Ehrhart polynomial of hypersimplices. It is proved that these polynomials have positive coefficients and we give a combinatorial formula for each of them. This settles a problem posed by Stanley and also proves that uniform matroids are Ehrhart positive, an important and yet unsolved particular case of a conjecture posed by De Loera et al. To this end, we introduce a new family of numbers that we call weighted Lah numbers and study some of their properties.