The Tutte polynomial via lattice point counting

TL;DR

This paper introduces a new polynomial for matroids and polymatroids derived from lattice point counting in Minkowski sums, linking combinatorial invariants with Ehrhart theory and extending known activity polynomials.

Contribution

It presents a novel polynomial that recovers the Tutte polynomial via lattice point enumeration, extending to polymatroids and connecting to existing activity polynomials.

Findings

Polynomial coefficients have alternating signs with combinatorial interpretation

The polynomial extends to polymatroids, unifying various invariants

Links Tutte polynomial with Ehrhart theory through lattice point counting

Abstract

We recover the Tutte polynomial of a matroid, up to change of coordinates, from an Ehrhart-style polynomial counting lattice points in the Minkowski sum of its base polytope and scalings of simplices. Our polynomial has coefficients of alternating sign with a combinatorial interpretation closely tied to the Dawson partition. Our definition extends in a straightforward way to polymatroids, and in this setting our polynomial has K\'alm\'an's internal and external activity polynomials as its univariate specialisations.