On Tutte polynomial expansion formulas in perspectives of matroids and oriented matroids

TL;DR

This paper introduces a new active partition concept for oriented matroid perspectives, leading to a simplified proof of Tutte polynomial expansion formulas and revealing structural insights into reorientations and activities.

Contribution

It generalizes previous constructions by Las Vergnas, providing a novel proof technique for Tutte polynomial expansions in matroids and oriented matroids.

Findings

Active partition defines equivalence classes of reorientations.

Short proof of a 4-variable Tutte polynomial expansion formula.

Extension to 5-variable expansion using subset activities.

Abstract

We introduce the active partition of the ground set of an oriented matroid perspective (or quotient, or strong map) on a linearly ordered ground set. The reorientations obtained by arbitrarily reorienting parts of the active partition share the same active partition. This yields an equivalence relation for the set of reorientations of an oriented matroid perspective, whose classes are enumerated by coefficients of the Tutte polynomial, and a remarkable partition of the set of reorientations into boolean lattices, from which we get a short direct proof of a 4-variable expansion formula for the Tutte polynomial in terms of orientation activities. This formula was given in the last unpublished preprint by Michel Las Vergnas; the above equivalence relation and notion of active partition generalize a former construction in oriented matroids by Michel Las Vergnas and the author; and the possibility of such a proof technique in perspectives was announced in the aforementioned preprint. We also briefly highlight how the 5-variable expansion of the Tutte polynomial in terms of subset activities in matroid perspectives comes in a similar way from the known partition of the power set of the ground set into boolean lattices related to subset activities (and we complete the proof with a property which was missing in the literature). In particular, the paper applies to matroids and oriented matroids on a linearly ordered ground set.