Representing Random Permutations as the Product of Two Involutions
نویسندگان
چکیده
An involution is a permutation that is its own inverse. Given a permutation σ of [n], let Nn(σ) denote the number of ways to write σ as a product of two involutions of [n]. If we endow the symmetric groups Sn with uniform probability measures, then the random variables Nn are asymptotically lognormal. The proof is based upon the observation that, for most permutations σ, Nn(σ) can be well approximated by Bn(σ), the product of the cycle lengths of σ. Asymptotic lognormality of Nn can therefore be deduced from Erdős and Turán’s theorem that Bn is itself asymptotically lognormal.
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