Ehrhart Theory of Paving and Panhandle Matroids

TL;DR

This paper develops a systematic method to compute Ehrhart polynomials of paving matroids by relating their base polytopes to hypersimplices and lattice path matroids, extending previous work and exploring Ehrhart positivity.

Contribution

It introduces a new geometric construction for paving matroids' base polytopes, generalizes existing formulas, and applies these results to matroids from Steiner systems and projective planes.

Findings

Ehrhart polynomials of paving matroids can be derived from hypersimplex slices.

Evidence suggests panhandle matroids are Ehrhart positive.

Ehrhart polynomials depend only on design parameters for certain matroids.

Abstract

We show that the base polytope $P_M$ of any paving matroid $M$ can be systematically obtained from a hypersimplex by slicing off certain subpolytopes, namely base polytopes of lattice path matroids corresponding to panhandle-shaped Ferrers diagrams. We calculate the Ehrhart polynomials of these matroids and consequently write down the Ehrhart polynomial of $P_M$, starting with Katzman's formula for the Ehrhart polynomial of a hypersimplex. The method builds on and generalizes Ferroni's work on sparse paving matroids. Combinatorially, our construction corresponds to constructing a uniform matroid from a paving matroid by iterating the operation of stressed-hyperplane relaxation introduced by Ferroni, Nasr, and Vecchi, which generalizes the standard matroid-theoretic notion of circuit-hyperplane relaxation. We present evidence that panhandle matroids are Ehrhart positive and describe a conjectured combinatorial formula involving chain forests and Eulerian numbers from which Ehrhart positivity of panhandle matroids will follow. As an application of the main result, we calculate the Ehrhart polynomials of matroids associated with Steiner systems and finite projective planes, and show that they depend only on their design-theoretic parameters: for example, while projective planes of the same order need not have isomorphic matroids, their base polytopes must be Ehrhart equivalent.