Involution factorizations of Ewens random permutations

TL;DR

This paper studies the distribution of involution factorizations of permutations under Ewens measures, showing asymptotic lognormality and phase transitions, extending previous results from uniform permutations to a broader class.

Contribution

It generalizes the asymptotic distribution results of involution factorizations from uniform permutations to Ewens measures with arbitrary positive parameter.

Findings

 invol is asymptotically lognormal for Ewens measures.

Identifies a phase transition at =1 in the behavior of invol.

Provides convergence rates and functional refinements for the Gaussian limit.

Abstract

An involution is a bijection that is its own inverse. Given a permutation $\sigma$ of $[n],$ let $\mathsf{invol}(\sigma)$ denote the number of ways $\sigma$ can be expressed as a composition of two involutions of $[n].$ We prove that the statistic $\mathsf{invol}$ is asymptotically lognormal when the symmetric groups $\mathfrak{S}_n$ are each equipped with Ewens Sampling Formula probability measures of some fixed positive parameter $\theta.$ This paper strengthens and generalizes previously determined results about the limiting distribution of $\log(\mathsf{invol})$ for uniform random permutations, i.e. the specific case of $\theta = 1$. We also investigate the first two moments of $\mathsf{invol}$ itself, detailing the phase transition in asymptotic behavior at $\theta = 1,$ and provide a functional refinement and a convergence rate for the Gaussian limit law which is demonstrably optimal when $\theta = 1.$