Involution factorizations of Ewens random permutations

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 $\theta = 1.$