Linear relations for a generalized Tutte polynomial

TL;DR

This paper extends Brylawski's linear relations for the Tutte polynomial to a broader class of combinatorial structures including greedoids and antimatroids, deriving new identities and relations.

Contribution

It generalizes linear relations of the Tutte polynomial to greedoids and antimatroids, introducing new identities and a novel relation for matroids.

Findings

Derived new linear identities for antimatroids, trees, posets, and chordal graphs.

Extended Brylawski's relations to a broader class of combinatorial objects.

Provided a new linear relation for matroids implied by existing identities.

Abstract

Brylawski proved the coefficients of the Tutte polynomial of a matroid satisfy a set of linear relations. We extend these relations to a generalization of the Tutte polynomial that includes greedoids and antimatroids. This leads to families of new identities for antimatroids, including trees, posets, chordal graphs and finite point sets in $\mathbb{R}^n$. It also gives a "new" linear relation for matroids that is implied by Brylawski's identities.