Counting Free Abelian Actions

TL;DR

This paper provides a general solution for counting commuting r-tuples in symmetric groups, extending known results for specific cases to arbitrary r, and confirming conjectures for r=3.

Contribution

It introduces a comprehensive formula for counting commuting r-tuples in S_n for any r, generalizing previous special cases and resolving a conjecture for r=3.

Findings

Derived a general formula for |Hom(Z^r,S_n)| for all r

Confirmed the conjecture for r=3 by Britnell

Extended known cases r=1,2 to arbitrary r

Abstract

We consider the problem of counting commuting r-tuples of elements of the symmetric group S_n, i.e. computing |Hom(Z^r,S_n)|. The cases r=1,2 are well-known; a product formula for the case r=3 was conjectured by Adams-Watters and later proved by Britnell. In this note we solve the problem for arbitrary r.