On the compatible sets expansion of the Tutte polynomial

TL;DR

This paper explores the relationship between a new compatible sets expansion formula for the Tutte polynomial and the classical internal-external activities formula, extending it to matroid perspectives and providing a direct proof of the bijection.

Contribution

It establishes a connection between the compatible sets expansion and activities expansion for the Tutte polynomial, extending the formula to matroid perspectives with a direct proof approach.

Findings

Derived a generalized expansion formula for Las Vergnas's trivariate Tutte polynomial.

Established a bijection between compatible sets and activities in matroid perspectives.

Provided a direct proof of the expansion formula using contraction-deletion relations.

Abstract

Kochol (2021) gave a new expansion formula for the Tutte polynomial of a matroid using the notion of \emph{compatible sets}, and asked how this expansion relates to the internal-external activities formula. Here, we provide an answer, which is obtained as a special case of a generalized version of the expansion formula to Las Vergnas's trivariate Tutte polynomials of matroid perspectives. The same generalization to matroid perspectives and bijection with activities have been independently proven by Kochol (2022 and 2023) in parallel with this work, but using different methods. Kochol proves both results recursively using the contraction-deletion relations, whereas we give a more direct proof of the bijection and use that to deduce the compatible sets expansion formula from Las Vergnas's activities expansion.