Hypersimplices are Ehrhart positive

2019/11/30 by Luis Ferroni · 2 citations
Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Advanced Mathematical Theories and Applications #Combinatorics #Conjecture #Discrete mathematics #Mathematics #Matroid #math.CO

paper · pdf · doi:10.1016/j.jcta.2020.105365

10 pages, 2 tables

arxiv created 2020/11/12 · openalex publication_date 2020/11/19 · arxiv updated 2020/11/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

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.

Citations

• A Monotonicity Property of h-vectors and h*-vectors
• Ehrhart Polynomials of Matroid Polytopes and Polymatroids
• Computing the Continuous Discretely
• Conjectures for Ehrhart h^*-vectors of Hypersimplices and Dilated Simplices
• Generalized permutahedra: Minkowski linear functionals and Ehrhart positivity

Cited by

• The Ehrhart polynomial of a matroid specializes to the beta invariant
• On the Ehrhart Polynomial of Minimal Matroids