The Tutte Polynomial of a Morphism of Matroids 5. Derivatives as Generating Functions of Tutte Activities

TL;DR

This paper demonstrates that derivatives of the Tutte polynomial in ordered matroids serve as generating functions for activities of specific subsets, extending to the 3-variable polynomial in matroid perspectives.

Contribution

It establishes a novel link between derivatives of the Tutte polynomial and generating functions of activities, generalizing to matroid perspectives.

Findings

Derivatives of the Tutte polynomial encode activities of subsets.

The property extends to the 3-variable Tutte polynomial of matroid perspectives.

Provides a new combinatorial interpretation of polynomial derivatives.

Abstract

We show that in an ordered matroid the partial derivative \partial^{p+q}t/\partialx^p\partialyq of the Tutte polynomial is p!q! times the generating function of activities of subsets with corank p and nullity q. More generally, this property holds for the 3-variable Tutte polynomial of a matroid perspective.