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Dissertation
Factoring Permutations into the Product of Two Involutions: A Probabilistic, Combinatorial, and Analytic Approach
Doctor of Philosophy (Ph.D.), Drexel University
Jun 2017
DOI:
https://doi.org/10.17918/etd-7344
Abstract
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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Details
- Title
- Factoring Permutations into the Product of Two Involutions
- Creators
- Charles Burnette - DU
- Contributors
- Eric J. Schmutz (Advisor) - Drexel University (1970-)
- Awarding Institution
- Drexel University
- Degree Awarded
- Doctor of Philosophy (Ph.D.)
- Publisher
- Drexel University; Philadelphia, Pennsylvania
- Number of pages
- vii, 60 pages
- Resource Type
- Dissertation
- Language
- English
- Academic Unit
- College of Arts and Sciences; Drexel University; Mathematics
- Other Identifier
- 7344; 991014632685504721