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Factoring permutations into the product of two involutions: a probabilistic, combinatorial, and analytic approach
Dissertation   Open access

Factoring permutations into the product of two involutions: a probabilistic, combinatorial, and analytic approach

Charles Burnette
Doctor of Philosophy (Ph.D.), Drexel University
Jun 2017
DOI:
https://doi.org/10.17918/etd-7344
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Abstract

Mathematics
An involution is a permutation that is its own inverse. Given a permutation [sigma] of [n], let N_n([sigma]) denote the number of ways to write [sigma] as a product of two involutions. The random variables N_n are asymptotically lognormal when the symmetric groups S_n are equipped with uniform probability measures, in particular, and more generally, Ewens measures of some fixed parameter [theta] > 0. The proof is based upon the observation that, for most permutations [sigma], the number of involution factorizations N_n([sigma]) can be well-approximated by B_n([sigma]), the product of the cycle lengths of [sigma]. The asymptotic lognormality of N_n can therefore be deduced from Erdős and Turán's theorem that B_n is itself asymptotically lognormal. We then briefly consider fixed-point free involution factorizations. A necessary and sufficient condition for a permutation to be the composition of two fixed-point free involutions is for it to have an even number of k-cycles, k = 1, 2, ... Through a combination of singularity analysis, the method of moments, and an appeal to the Shepp-Lloyd model for random permutations, the asymptotic enumeration and cycle structure of random permutations admitting fixed-point free involution factorizations are calculated.

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