The $\bar\gamma$-frame for Tutte polynomials of matroids

TL;DR

This paper introduces a new basis called the $ar ext{ extgamma}$-frame for expressing Tutte polynomials of matroids, providing explicit formulas and non-negative integer coefficient expansions.

Contribution

It defines a novel spanning set for Tutte polynomials of matroids and derives explicit formulas for its elements, enabling structured polynomial expansions.

Findings

Provides explicit formulas for the $ar ext{ extgamma}$-frame elements.

Shows every Tutte polynomial can be expanded with non-negative integer coefficients.

Offers a combinatorial interpretation of expansion coefficients as sums over flats.

Abstract

Specializing the $\gamma$-basis for the vector space $\mathcal{G}(n,r)$ spanned by the set of symbols on bit sequences with $r$ $1$'s and $n-r$ $0$'s, we obtain a frame or spanning set for the vector space $\mathcal{T}(n,r)$ spanned by Tutte polynomials of matroids having rank $r$ and size $n$. Every Tutte polynomial can be expanded as a linear combination with non-negative integer coefficients of elements in this frame. We give explicit formulas for the elements in this frame. These formulas combine to give an expansion of the Tutte polynomial with coefficients obtained by summing numerical invariants over all flats with a given rank and size.