Ehrhart theory of paving and panhandle matroids

Derek Hanely, Jeremy L. Martin, Daniel McGinnis, Dane Miyata, George D. Nasr, Andrés R. Vindas-Meléndez, Mei Yin

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1 Scopus citations

Abstract

We show that the base polytope PM 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 PM, 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.

Original languageEnglish (US)
Pages (from-to)501-526
Number of pages26
JournalAdvances in Geometry
Volume23
Issue number4
DOIs
StatePublished - Oct 1 2023

All Science Journal Classification (ASJC) codes

  • Geometry and Topology

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