Symmetric inclusion-exclusion

TL;DR

This paper explores a symmetric form of the inclusion-exclusion principle, analyzing combinatorial and probabilistic cases, especially those related to the Polya-Eggenberger urn model where functions depend only on set size.

Contribution

It introduces and studies a symmetric inclusion-exclusion framework with applications to combinatorial and probabilistic models, including urn processes.

Findings

Identifies conditions for symmetric inclusion-exclusion to hold

Analyzes cases where functions depend only on set cardinality

Connects symmetric inclusion-exclusion to urn models

Abstract

One form of the inclusion-exclusion principle asserts that if A and B are functions of finite sets then A(S) is the sum of B(T) over all subsets T of S if and only if B(S) is the sum of (-1)^|S-T| A(T) over all subsets T of S. If we replace B(S) with (-1)^|S| B(S), we get a symmetric form of inclusion-exclusion: A(S) is the sum of (-1)^|T| B(T) over all subsets T of S if and only if B(S) is the sum of (-1)^|T| A(T) over all subsets T of S. We study instances of symmetric inclusion-exclusion in which the functions A and B have combinatorial or probabilistic interpretations. In particular, we study cases related to the Polya-Eggenberger urn model in which A(S) and B(S) depend only on the cardinality of S.