Involution factorizations of Ewens random permutations
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866918128005414912 |
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| author | Burnette, Charles |
| author_facet | Burnette, Charles |
| contents | An involution is a bijection that is its own inverse. Given a permutation $σ$ of $[n],$ let $\mathsf{invol}(σ)$ denote the number of ways $σ$ 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 $θ.$ 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 $θ= 1$. We also investigate the first two moments of $\mathsf{invol}$ 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 $θ= 1.$ |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2105_12695 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Involution factorizations of Ewens random permutations Burnette, Charles Combinatorics Probability 05A05, 05A16, 60C05 An involution is a bijection that is its own inverse. Given a permutation $σ$ of $[n],$ let $\mathsf{invol}(σ)$ denote the number of ways $σ$ 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 $θ.$ 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 $θ= 1$. We also investigate the first two moments of $\mathsf{invol}$ 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 $θ= 1.$ |
| title | Involution factorizations of Ewens random permutations |
| topic | Combinatorics Probability 05A05, 05A16, 60C05 |
| url | https://arxiv.org/abs/2105.12695 |