Journal article
Representing Random Permutations as the Product of Two Involutions
21 Jul 2015
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 Erd\H{o}s and Tur\'{a}n's
theorem that $\mathbf{B}_{n}$ is itself asymptotically lognormal.
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Details
- Title
- Representing Random Permutations as the Product of Two Involutions
- Creators
- Charles BurnetteEric Schmutz
- Resource Type
- Journal article
- Language
- English
- Academic Unit
- Mathematics
- Identifiers
- 991019173767304721