Relations between M\"obius and coboundary polynomial

TL;DR

This paper investigates the relationship between the M"obius and coboundary polynomials of matroids, identifying specific cases where they determine each other and providing constructions for these scenarios.

Contribution

It introduces conditions under which the M"obius polynomials of a matroid and its dual uniquely determine the coboundary polynomial, along with explicit constructions.

Findings

In certain cases, the M"obius polynomials determine the coboundary polynomial.

Provides two constructions to derive the coboundary polynomial from M"obius polynomials.

Clarifies the relationship between these polynomials in specific matroid classes.

Abstract

It is known that, in general, the coboundary polynomial and the M\"obius polynomial of a matroid do not determine each other. Less is known about more specific cases. In this paper, we will try to answer if it is possible that the M\"obius polynomial of a matroid, together with the M\"obius polynomial of the dual matroid, define the coboundary polynomial of the matroid. In some cases, the answer is affirmative, and we will give two constructions to determine the coboundary polynomial in these cases.