University of Pennsylvania ScholarlyCommons Publicly Accessible Penn Dissertations 2010 Profiles of Large Combinatorial Structures Michael T. Lugo University of Pennsylvania, [email protected] Follow this and additional works at: https://repository.upenn.edu/edissertations Part of the Discrete Mathematics and Combinatorics Commons, and the Probability Commons Recommended Citation Lugo, Michael T., "Profiles of Large Combinatorial Structures" (2010). Publicly Accessible Penn Dissertations. 127. https://repository.upenn.edu/edissertations/127 This paper is posted at ScholarlyCommons. https://repository.upenn.edu/edissertations/127 For more information, please contact [email protected]. Profiles of Large Combinatorial Structures Abstract We derive limit laws for random combinatorial structures using singularity analysis of generating functions. We begin with a study of the Boltzmann samplers of Flajolet and collaborators, a useful method for generating large discrete structures at random which is useful both for providing intuition and conjecture and as a possible proof technique. We then apply generating functions and Boltzmann samplers to three main classes of objects: permutations with weighted cycles, involutions, and integer partitions. Random permutations in which each cycle carries a multiplicative weight $\sigma$ have probability $(1-\gamma)^\sigma$ of having a random element be in a cycle of length longer than $\gamma n$; this limit law also holds for cycles carrying multiplicative weights depending on their length and averaging $\sigma$. Such permutations have number of cycles asymptotically normally distributed with mean and variance $\sim \sigma \log n$. For permutations with weights $\sigma_k = 1/k$ or $\sigma_k = k$, other limit laws are found; the prior have finitely many cycles in expectation, the latter around $\sqrt{n}$. Compositions of uniformly chosen involutions of $[n]$, on the other hand, have about $\sqrt{n}$ cycles on average. These can be modeled as modified 2-regular graphs. A composition of two random involutions in $S_n$ typically has about $n^{1/2}$ cycles, characteristically of length $n^{1/2}$. The number of factorizations of a random permutation into two involutions appears to be asymptotically lognormally distributed, which we prove for a closely related probabilistic model. We also consider connections to pattern avoidance, in particular to the distribution of the number of inversions in involutions. Last, we consider integer partitions. Various results on the shape of random partitions are simple to prove in the Boltzmann model. We give a (conjecturally tight) asymptotic bound on the number of partitions $p_M(n)$ in which all part multiplicities lie in some fixed set $n$, and explore when that asymptotic form satisfies $\log p_M(n) \sim \pi\sqrt{Cn}$ for rational $C$. Finally we give probabilistic interpretations of various pairs of partition identities and study the Boltzmann model of a family of random objects interpolating between partitions and overpartitions. Degree Type Dissertation Degree Name Doctor of Philosophy (PhD) Graduate Group Mathematics First Advisor Robin Pemantle Keywords generating functions, random permutations, Boltzmann samplers, class multiplication problem, singularity analysis, integer partitions Subject Categories Discrete Mathematics and Combinatorics | Probability This dissertation is available at ScholarlyCommons: https://repository.upenn.edu/edissertations/127 PROFILES OF LARGE COMBINATORIAL STRUCTURES Michael T. Lugo A DISSERTATION in Mathematics Presented to the Faculties of the University of Pennsylvania in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy 2010 Supervisor of Dissertation Robin Pemantle, Merriam Term Professor of Mathematics Graduate Group Chairperson Tony Pantev, Professor of Mathematics Dissertation Committee: Jason Bandlow, Lecturer of Mathematics Philip Gressman, Assistant Professor of Mathematics Robin Pemantle, Merriam Term Professor of Mathematics Acknowledgements First I must thank Robin Pemantle, my advisor. I have greatly enjoyed working with him for the past five years. His advice and encouragement has been crucial to my development as a mathematician. He has given me wide latitude to pursue my own interests but at the same time steered me towards problems that other people might actually care about. He has known when to push me and when to back off, when to let me wander and when to bring me back in. He has explained to me how probabilists think. His door has always been open, even if I didn’t take advantage of that as much as I could have. I am honored to be his student. The various teachers I’ve had in combinatorics and probability courses taught me, if not everything they know, at least large portions thereof. These are Miklos Bona, Count von Count, David Galvin, Marko Petkovsek, Richard Stanley, J. Michael Steele, Balint Virag, Mark Ward, and Herb Wilf. In particular Mark Ward taught a course in analytic combinatorics from Flajolet and Sedgewick’s book in the fall of 2006, which introduced me to the power and beauty of analytic combinatorics and which pointed me in the direction which eventually led to this thesis. (I had also ii just moved at the beginning of the semester and didn’t have Internet access for the first few weeks. I believe this is not a coincidence.) At MIT, Michael Artin’s spring 2004 Project Laboratory in Mathematics (18.821) introduced me to a “toy” version of mathematical research, and Igor Pak was my research advisor for an undergraduate research project the following summer. It was in this period that I realized that I had what it takes to do mathematical research. Finally, Jason Bandlow and Philip Gress- man served ably on my thesis committee, and Angela Gibney advised me through many of the difficulties of the first year of graduate school. As everyone knows, you can’t let mathematicians manage themselves. The staff of the Penn mathematics department – Janet Burns, Monica Pallanti, Paula Scarbor- ough, and Robin Toney – have kept the department moving and been friendly faces who have never been disappointed at me for not having made mathematical progress. Henry Benjamin also deserves thanks, for keeping the computers running. My time at Penn has been enhanced by my fellow graduate students, both for their moral support and encouragement and for their willingness to listen to my mathematical ideas. In particular I’d like to acknowledge Andrew Bressler, Ricky Der, Tim de Vries, Shanshan Ding, Jonathan Kariv, Paul Levande, Alexa Mater, Julius Poh, Andrew Rupinski, Benjamin Schak, Charles Siegel, Michael Thompson, and Mirko Visontai. I had the pleasure of giving a talk on much of this material at the Cornell Univer- sity mathematics department in February of 2010. I’d like to thank Rick Durrett for iii the invitation. Much of the material on partitions in this thesis first began to take shape during the Cornell Probability Summer School in 2009. I’ve also given talks on various portions of this thesis in seminars at Penn, both the graduate student combi- natorics seminar and the “grown-up” combinatorics/probability seminar; I thank the organizers of these seminars for letting me speak and giving me reasonably low-stress forums for shaping these ideas. The material of Sections 4.1 through 4.4 was previously published in the Electronic Journal of Combinatorics. I thank the anonymous referee of that paper for remarks concerning the proof of Theorem 4.2.7. I also thank Mirko Visontai for pointing out that Theorem 4.9.5 was proven in Stanley’s text; in a version of Section 4.9 that I previously circulated I gave a (somewhat unwieldy) proof. Graduate school takes a long time and is a very stressful experience. I’ve had the pleasure of being able to attend graduate school close to my family and having a family that I actually want to be close to. I would like to apologize to my cousin, John DeCaro for not inventing the “Italian restaurant process” or naming any object in this thesis after a type of pasta. I thank my parents, Janis and Albert Lugo. Long ago they tried to teach me what square numbers were by rearranging pennies on a kitchen table; I asked, innocently, if there were “twiangle numbers”. There are, of course; this was just an early example of a seemingly endless stream of questions that they endured for the most part with good humor. They have supported me through good times and bad and have always provided a home for me. This thesis iv is as much their achievement as mine. Finally, my grandmother, Josephine DeCaro, suffers from Alzheimer’s disease, and can no longer fully appreciate the importance of this moment in my life. But I know she would be proud. v ABSTRACT PROFILES OF LARGE COMBINATORIAL STRUCTURES Michael T. Lugo Robin Pemantle, Advisor We derive limit laws for random combinatorial structures using singularity anal- ysis of generating functions. We begin with a study of the Boltzmann samplers of Flajolet and collaborators, a useful method for generating large discrete structures at random which is useful both for providing intuition and conjecture and as a possible proof technique. We then apply generating functions and Boltzmann samplers to three main classes of objects: permutations with weighted cycles, involutions, and integer partitions. Random permutations in which each cycle carries a multiplica- tive weight σ have probability (1 − γ)σ of having a random element be in a cycle of length longer than γn; this limit law also holds for cycles carrying multiplicative weights depending on their length and averaging σ. Such permutations have num- ber of cycles asymptotically normally distributed with mean and variance ∼ σ log n. For permutations with weights σk = 1/k or σk = k, other limit laws are found; the √ prior have finitely many cycles in expectation, the latter around n. Compositions √ of uniformly chosen involutions of [n], on the other hand, have about n cycles on average. These can be modeled as modified 2-regular graphs. A composition of two 1/2 random involutions in Sn typically has about n cycles, characteristically of length vi n1/2.