2021/05/26 by Charles Burnette, Burnette, Charles
Computer Science · Mathematics · #05A05 #05A16 #60C05 #Advanced Combinatorial Mathematics #Bayesian Methods and Mixture Models #Combinatorics (math.CO) #FOS: Mathematics #Probability (math.PR) #Stochastic processes and statistical mechanics
paper · doi:10.48550/arxiv.2105.12695
openalex publication_date 2021/05/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
An involution is a bijection that is its own inverse. Given a permutation σ of [n], let invol(σ) denote the number of ways σ can be expressed as a composition of two involutions of [n]. We prove that the statistic invol is asymptotically lognormal when the symmetric groups \mathfrakSn are each equipped with Ewens Sampling Formula probability measures of some fixed positive parameter θ. This paper strengthens and generalizes previously determined results about the limiting distribution of log(invol) for uniform random permutations, i.e. the specific case of θ= 1. We also investigate the first two moments of invol itself, detailing the phase transition in asymptotic behavior at θ= 1, and provide a functional refinement and a convergence rate for the Gaussian limit law which is demonstrably optimal when θ= 1.