A bijection for tuples of commuting permutations and a log-concavity conjecture

TL;DR

This paper introduces a new elementary bijection to prove a formula for counting tuples of commuting permutations with a fixed number of orbits, and explores a related log-concavity conjecture extending previous work.

Contribution

It provides a self-contained bijective proof of an explicit formula for $A(\,ell,n,k)$ and investigates a new log-concavity conjecture generalizing prior results.

Findings

Explicit bijection for counting tuples of commuting permutations.

Proof of a formula for $A(\,ell,n,k)$ based on this bijection.

Evidence supporting the log-concavity conjecture.

Abstract

Let $A(\ell,n,k)$ denote the number of $\ell$-tuples of commuting permutations of $n$ elements whose permutation action results in exactly $k$ orbits or connected components. We provide a new proof of an explicit formula for $A(\ell,n,k)$ which is essentially due to Bryan and Fulman, in their work on orbifold higher equivariant Euler characteristics. Our proof is self-contained, elementary, and relies on the construction of an explicit bijection, in order to perform the $\ell+1\rightarrow \ell$ reduction. We also investigate a conjecture by the first author, regarding the log-concavity of $A(\ell,n,k)$ with respect to $k$. The conjecture generalizes a previous one by Heim and Neuhauser related to the Nekrasov-Okounkov formula.