Representing Random Permutations as the Product of Two Involutions

Abstract

An involution is a permutation that is its own inverse. Given a permutation $\sigma$ of $[n],$ let ${\mathbf{N}}_n(\sigma )$ denote the number of ways to write $\sigma$ as a product of two involutions of $[n].$ If we endow the symmetric groups $S_n$ with uniform probability measures, then the random variables ${\mathbf{N}}_n$ are asymptotically lognormal. The proof is based upon the observation that, for most permutations $\sigma$, ${\mathbf{N}}_n(\sigma )$ can be well approximated by ${\mathbf{B}}_n(\sigma ),$ the product of the cycle lengths of $\sigma$. Asymptotic lognormality of ${\mathbf{N}}_n$ can therefore be deduced from Erdős and Turán's theorem that ${\mathbf{B}}_n$ is itself asymptotically lognormal.

10 pages

References in corpus (1)

• The cycle structure of compositions of random involutions

Cited by in corpus (1)

• Involution factorizations of Ewens random permutations