Hypersimplices are Ehrhart Positive
arXiv:1911.10146 · doi:10.1016/j.jcta.2020.105365
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.
10 pages, 2 tables