Representing Random Permutations as the Product of Two Involutions

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 Erd\H{o}s and Tur\'{a}n's theorem that $\mathbf{B}_{n}$ is itself asymptotically lognormal.