On the number of cycles in commutators of random permutations

TL;DR

This paper explores the relationship between non-Hermitian random matrix statistics and the cycle structure of certain random permutations, providing explicit formulas for cycle counts in commutators.

Contribution

It introduces explicit formulas for the generating functions of cycle counts in commutators of specific random permutations, linking permutation statistics to random matrix theory.

Findings

Derived explicit formulas for generating functions of cycle counts

Established connections between permutation cycles and non-Hermitian matrix statistics

Analyzed cycle structures in commutators involving various permutation types

Abstract

We present general links between statistics of non-Hermitian random matrices and the distribution of the number of cycles of some specific random permutations. In particular, we derive explicit formulas for the generating functions of the number of cycles in the commutator $[\sigma,\tau] = \sigma \tau \sigma^{-1} \tau^{-1}$ where $\sigma$ is uniformly distributed, and $\tau$ is either one cycle, the product of two cycles of same size, or the product of many transpositions.