Uniform Matroids are Ehrhart Positive

Luis Ferroni

Published 2019-11-22Version 1

In [1] De Loera et al. conjectured that the Ehrhart polynomial of the basis polytope of a matroid has positive coefficients. We prove this conjecture for all uniform matroids. In other words, we prove that every hypersimplex is Ehrhart positive. In order to do that, we introduce the notion of weighted Lah numbers and study some of their properties. Then we provide a formula for the coefficients of the Ehrhart polynomial of a hypersimplex in terms of these numbers.