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Ehrhart Bounds for Panhandle and Paving Matroids Through Enumeration of Chain Forests

Danai Deligeorgaki, Daniel McGinnis, Andrés R. Vindas-Meléndez

TL;DR

This work resolves key questions about Ehrhart positivity for two matroid-polytope families by recasting the problem into pure enumerative combinatorics. Central to the approach is a novel bijection between valued $A$-distinguished ordered chain forests and ordinary chain forests, together with a sign-reversing involution that cancels most terms in a crucial alternating sum, leaving a positive-leading contribution that proves positivity for panhandle matroids. The authors extend the framework to paving matroids, establishing an Ehrhart-coefficient upper bound relative to the uniform matroid via relaxations of stressed hyperplanes and nonnegative-coefficient polynomials. The results strengthen connections between matroid polytopes, alcoved and polypositroid structures, and broader conjectures about Ehrhart positivity in related matroid families, demonstrating a robust combinatorial toolkit for Ehrhart questions in this domain.

Abstract

Panhandle matroids are a specific family of lattice-path matroids corresponding to panhandle-shaped Ferrers diagrams. Their matroid polytopes are the subpolytopes carved from a hypersimplex to form matroid polytopes of paving matroids. It has been an active area of research to determine which families of matroid polytopes are Ehrhart positive. We prove Ehrhart positivity for panhandle matroid polytopes, thus confirming a conjecture of Hanely, Martin, McGinnis, Miyata, Nasr, Vindas-Meléndez, and Yin (2023). Another standing conjecture posed by Ferroni (2022) asserts that the coefficients of the Ehrhart polynomial of a connected matroid are bounded above by those of the corresponding uniform matroid. We prove Ferroni's conjecture for paving matroids -- a class conjectured to asymptotically contain all matroids. These results follow from purely enumerative statements, the main one being conjectured by Hanely et. al concerning the enumeration of a certain class of ordered chain forests.

Ehrhart Bounds for Panhandle and Paving Matroids Through Enumeration of Chain Forests

TL;DR

This work resolves key questions about Ehrhart positivity for two matroid-polytope families by recasting the problem into pure enumerative combinatorics. Central to the approach is a novel bijection between valued -distinguished ordered chain forests and ordinary chain forests, together with a sign-reversing involution that cancels most terms in a crucial alternating sum, leaving a positive-leading contribution that proves positivity for panhandle matroids. The authors extend the framework to paving matroids, establishing an Ehrhart-coefficient upper bound relative to the uniform matroid via relaxations of stressed hyperplanes and nonnegative-coefficient polynomials. The results strengthen connections between matroid polytopes, alcoved and polypositroid structures, and broader conjectures about Ehrhart positivity in related matroid families, demonstrating a robust combinatorial toolkit for Ehrhart questions in this domain.

Abstract

Panhandle matroids are a specific family of lattice-path matroids corresponding to panhandle-shaped Ferrers diagrams. Their matroid polytopes are the subpolytopes carved from a hypersimplex to form matroid polytopes of paving matroids. It has been an active area of research to determine which families of matroid polytopes are Ehrhart positive. We prove Ehrhart positivity for panhandle matroid polytopes, thus confirming a conjecture of Hanely, Martin, McGinnis, Miyata, Nasr, Vindas-Meléndez, and Yin (2023). Another standing conjecture posed by Ferroni (2022) asserts that the coefficients of the Ehrhart polynomial of a connected matroid are bounded above by those of the corresponding uniform matroid. We prove Ferroni's conjecture for paving matroids -- a class conjectured to asymptotically contain all matroids. These results follow from purely enumerative statements, the main one being conjectured by Hanely et. al concerning the enumeration of a certain class of ordered chain forests.
Paper Structure (7 sections, 21 theorems, 57 equations, 1 table, 1 algorithm)

This paper contains 7 sections, 21 theorems, 57 equations, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Toward Ehrhart positivity: Combinatorial objects
  3. A bijection between ordered chain forests
  4. An algorithm defined on valued -distinguished ordered chain forests.
  5. Properties of the map
  6. Panhandle matroids are Ehrhart positive
  7. Ehrhart-coefficient upper-bound conjecture for paving matroids

Key Result

Theorem 1.1

Let $1\leq r\leq s< n$. The Ehrhart polynomial of $\mathcal{P}_{\mathop{\mathrm{\mathsf{Pan}}}\nolimits_{r,s,n}}$ can be written as where

Theorems & Definitions (54)

  • Theorem 1.1: Corollary 5.4 in hanelyetal
  • Conjecture 1.2: Conjecture 6.9 in hanelyetal
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5: ferroniPositivehanelyetal
  • Definition 2.6
  • ...and 44 more