View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Drexel Libraries E-Repository and Archives Factoring Permutations into the Product of Two Involutions: A Probabilistic, Combinatorial, and Analytic Approach A Thesis Submitted to the Faculty of Drexel University by Charles Burnette in partial fulfillment of the requirements for the degree of Doctor of Philosophy June 2017 c Copyright 2017 Charles Burnette. All Rights Reserved ii Acknowledgments First I must thank my thesis advisor, Eric Schmutz, for his exceptional mentorship. It has been a pleasure working with him for the past five years. His advice and guidance helped mold me as a mathematician. He granted me the freedom and independence to pursue my own interests, but was also a great source of encouragement and advice whenever I struggled. He has always made himself available for help, even though I did not capitalize on this opportunity as much as I should have. His patience and friendliness helped alleviate much of the stress of navigating graduate school, conferences, and the job market. I am very honored to be his student. I have been extremely fortunate to experience a quality education in combinatorics, analysis, and probability. Courses that I took under David Ambrose, Robert Boyer, Simon Foucart, Peter Freyd, Louis Friedler, Pawe ll Hitczenko, Ryan Kaliszewski, Dmitry Kaliuzhniy-Verbovetsky, Jen- nifer Morse, Shari Moskow, Alexander Nagel, Fedor Nazarov, Ron Perline, Jean-Pierre Rosay, Eric Schmutz, Andreas Seeger, Paul Terwiliger, and Ned Wolff shaped most of my knowledge in these areas of mathematics. Other mathematicians who have personally shared their deep insights on combinatorics and related mathematics with me on numerous occasions include Jonah Blasiak, Jim Haglund, Huilan Li, Victor Moll, Ken Ono, Greta Panova, Robin Pemantle, Mark Ward, Hugo Wo- erdeman, Dirk Zeindler, and many others who have influenced, either directly or indirectly, my past and present research themes and directions. Mark Ward especially inspired me to try and master analytic combinatorics. After I attended his mini-lectures during the 2nd Lake Michigan Workshop on Combinatorics and Graph Theory at Notre Dame, I was denuded of the misgivings I previously had about learning this subject. I was hooked. I also wish to thank the administrative staff of the math department { Paige Chmielewski, Kenn Hemphill, Gene Phan, Sobha Phillips, and Amy Tiernan { who work behind-the-scenes to make sure that the department runs smoothly, that the technology runs properly, and that I get paid/reimbursed. I would also like to thank the late Mindy Gilchrist, who used to be on the staff but iii passed away midway through my time at Drexel. We spoke together a lot, and she always wanted to know whether everything was going okay. I had a lot of fun these past five years mostly because of my fellow graduate students. I especially want to acknowledge Avi Dalal, Tim Faver, Tim Hayes, Mandy Lohss, Michael Minner, Dan Parry, Sarah Rody, Scott Rome, Pat Shields, David Sulon, Kelly Toppin, Chung Wong, and Trevor Zaleski for all of the times we hanged out together and exchanged ideas on everything from mathematics to sports, from game shows to Salty Bet. I also want to thank the organizers of the local seminars at Drexel, Penn, and Bryn Mawr at which I gave multiple talks for providing me an occasional platform. This also goes for all of the conferences that I've presented at. I have received so many helpful comments from people I don't know but decided to listen to me anyway. I have also received great suggestions from anonymous referees on the papers that I have either published or submitted for publication. These experiences really drive me to improve my exposition and oratory, and I'm grateful for it. I now want to take the time to acknowledge those who I feel are the main reasons why I even decided to go into mathematics in the first place. I greatly thank Louis Friedler, Carlos Ortiz, and Ned Wolf, my three math professors from Arcadia. I entered my undergrad as an English major without so much as a scintilla of interest in doing anything genuinely meaningful with my life. I enrolled in a Calculus I class on a whim. (As an English major, I didn't have to take it. Something at the level of Precalculus would have sufficed.) But I took Prof. Ortiz's calculus course anyway, and I didn't want to stop there. Next thing I know, I'm minoring in math. Next thing I know, I'm declaring a second major in math. Next thing I know, I'm ditching the English major and deciding to pursue a Ph.D. in math. (Apologies to the magnificent English professors that I had at Arcadia. I do occasionally feel like a traitor.) And this sequence of changes in my declared major is primarily because of these three people. They looked out for me more than any other teachers I've ever had up to that point. They were the first to ever tell me that I had the talent to make a living out of doing mathematics. I believed them. And I still do. Last but not least, I want to thank my parents, Barbara Burnette and Charles Burnette, Sr., iv for the limitless amount of love and support that they have given me. My mom has sacrificed so much to make sure that I had the chance and the means to pursue my ambitions. Even as a grown man now in his thirties, she's quick to look out for me. My dad was also very instrumental in my intellectual development. A degenerative eye disorder robbed him of his vision when I was very young. So he would have me read books and newspaper articles to him. As much as hated doing this at times, it instilled in me a thirst for knowledge. I finish with one last blanket statement of indebtedness to anyone important I may be forgetting about. I feel very privileged to have had so many people throughout my life invested in my success. Not many people get as many redos, mulligans, and second chances as I've gotten. I made it this far mostly because of the generosity, kindness, and mercy of others. So again, thank you to you all. Spread love. It's the Brooklyn way. { The Notorious B.I.G. v Table of Contents Abstract .............................................. vii 1. Introduction .......................................... 1 1.1 A Brief History of Analytic Combinatorics......................... 1 1.2 Statement of Results..................................... 2 2. Background Mathematics .................................. 5 2.1 The Generatingfunctionology of Permutations....................... 5 2.1.1 Generating Functions Associated with Permutations ................. 5 2.1.2 The Shepp-Lloyd Model................................. 8 2.2 Singularity Analysis ..................................... 9 2.2.1 The Gestalt of Analytic Combinatorics......................... 9 2.2.2 Wright's Expansions ................................... 11 2.2.3 Darboux's Method .................................... 11 2.2.4 A Flajolet-Odlyzko Transfer Theorem ......................... 13 2.2.5 Hybridization of Darboux's Method and the Transfer Theorem ........... 14 2.2.6 A Coefficient Growth Estimate ............................. 17 2.3 The Method of Moments................................... 17 3. Factoring Random Permutations ............................. 19 3.1 Preliminaries ......................................... 19 3.2 The Combinatorics Behind Involution Factorizations................... 20 3.3 Extremal Properties of Nn ................................. 24 3.4 Approximation by Bn .................................... 27 3.5 The Asymptotic Lognormality of Nn ............................ 29 4. Generalizing to the Ewens Sampling Formula .................... 31 4.1 What Is the Ewens Sampling Formula?........................... 31 vi 4.2 Mean Number of Involution Factorizations......................... 32 4.3 Approximation by Bn Revisited............................... 34 4.4 The Asymptotic Lognormality of Nn Revisited ...................... 37 4.5 The Threshold of θ = 1 ................................... 38 5. Permutations Admitting Fixed-Point Free Involution Factorizations . 40 5.1 Motivation .......................................... 40 5.2 Asymptotic Enumeration of Permissible Permutations .................. 41 ∗ 5.3 Extremal Properties of N2n ................................. 44 5.4 Cycle Structure of Permissible Permutations........................ 46 5.5 Concluding Remarks..................................... 49 Bibliography ............................................ 50 Appendix A: Alternative Derivation of (4.1.6) ...................... 54 ∗ Appendix B: Generating Function for E2n(N2n) ..................... 56 Appendix C: Alternative Proof of Theorem 5.4.1 .................... 57 Appendix D: Frequently Used Notation .......................... 58 Vita .................................................. 60 vii Abstract Factoring Permutations into the Product of Two Involutions: A Probabilistic, Combinatorial, and Analytic Approach Charles Burnette Eric Schmutz, Ph.D. 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. The random variables Nn are asymptotically lognormal when the symmetric groups Sn are equipped with uniform probability measures, in particular, and, more generally, Ewens measures of some fixed parameter θ > 0: The proof is based upon the observation that, for most permutations σ; the number of involution factorizations Nn(σ) can be well-approximated by Bn(σ); the product of the cycle lengths of