
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.
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Taxonomy
TopicsComputability, Logic, AI Algorithms · Cellular Automata and Applications · Advanced Topology and Set Theory
