Convolution-multiplication identities for Tutte polynomials of matroids

TL;DR

This paper presents a general convolution identity for Tutte polynomials of matroids, unifying and extending known identities with combinatorial and probabilistic interpretations.

Contribution

It introduces a broad multiplication-convolution identity for the rank generating polynomial, linking it to the Tutte polynomial through simple algebraic transformations.

Findings

Unified convolution identity for Tutte and rank generating polynomials

Derivation of multiple known identities as special cases

Provides combinatorial and probabilistic interpretations

Abstract

We give a general multiplication-convolution identity for the multivariate and bivariate rank generating polynomial of a matroid. The bivariate rank generating polynomial is transformable to and from the Tutte polynomial by simple algebraic operations. Several identities, almost all already known in some form, are specializations of this identity. Combinatorial or probabilistic interpretations are given for the specialized identities.