Inclusion-exclusion by ordering-free cancellation

TL;DR

This paper introduces an ordering-free cancellation method for inclusion-exclusion formulas, extending previous ordering-dependent techniques and reducing more terms, with applications to graph polynomials.

Contribution

The authors develop a novel cancellation approach for inclusion-exclusion that does not rely on any ordering, broadening the applicability of existing methods.

Findings

The new method generalizes all known ordering-based cancellations.

It reduces more terms in inclusion-exclusion formulas.

Applications include improved results on graph polynomials.

Abstract

Whitney's broken circuit theorem gives a graphical example to reduce the number of the terms in the sum of the inclusion-exclusion formula by a predicted cancellation. So far, the known cancellations for the formula strongly depend on the prescribed (linear or partial) ordering on the index set. We give a new cancellation method, which does not require any ordering on the index set. Our method extends all the `ordering-based' methods known in the literatures and in general reduces more terms. As examples, we use our method to improve some relevant results on graph polynomials.