A lattice point counting generalisation of the Tutte polynomial

TL;DR

This paper introduces a new polynomial for polymatroids that generalizes the Tutte polynomial, capturing similar combinatorial information through lattice point counts, and retains key properties when restricted to matroids.

Contribution

A novel polynomial for polymatroids is constructed, extending Tutte polynomial concepts via lattice point enumeration in Minkowski sums, with properties aligning with matroid cases.

Findings

Polynomial contains the same information as Tutte polynomial for matroids

Coefficients exhibit alternating signs with combinatorial interpretation

Method uses lattice point counts in Minkowski sums of polytopes

Abstract

The Tutte polynomial for matroids is not directly applicable to polymatroids. For instance, deletion-contraction properties do not hold. We construct a polynomial for polymatroids which behaves similarly to the Tutte polynomial of a matroid, and in fact contains the same information as the Tutte polynomial when we restrict to matroids. This polynomial is constructed using lattice point counts in the Minkowski sum of the base polytope of a polymatroid and scaled copies of the standard simplex. We also show that, in the matroid case, our polynomial has coefficients of alternating sign, with a combinatorial interpretation closely tied to the Dawson partition.