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Shuffling Decks with Repeated Card Values Shuffling Decks With Repeated Card Values by Mark A. Conger A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Mathematics) in The University of Michigan 2007 Doctoral Committee: Assistant Professor Divakar Viswanath, Chair Professor Sergey Fomin Professor John K. Riles Professor Bruce Sagan, Michigan State University c Mark A. Conger All Rights Reserved 2007 ACKNOWLEDGEMENTS This thesis has been more than 16 years in the making, so there are a lot of people to thank. Thanks to my friends and roommates Steve Blair and Paul Greiner, who managed to get me through my first year at Michigan (1990-91). As Steve said to me once, years later, “Together, we made a pretty good graduate student.” Thanks to Prof. Larry Wright at Williams for teaching me about Radix Sort. And thanks to everyone who shared their punch cards and punch card stories with me: John Remmers, Ken Josenhans, Thom Sterling, and Larry Mohr among others. Thanks to Profs. Bill Lenhart at Williams and Phil Hanlon at Michigan who taught me basic enumerative combinatorics, especially how to switch summation signs. Prof. Ollie Beaver taught me how to use confidence intervals, and Martin Hildebrand taught me the Central Limit Theorem. Pam Walpole and Pete Morasca taught me many things I use every day. Most of the programs used to test hypotheses and generate Monte Carlo data were written in C/C++. The thesis was typeset in LATEX using the PSTricks package, with a number of figures written directly in PostScript. Other figures, as well as the glossary and notation pages, were generated from scripts by Perl programs. Certain ii tables were created by an XSLT stylesheet. MAPLE programs were used when large rational results were required. So thank you to Brian Kernighan, Dennis Ritchie, Bjarne Stroustrup, Donald Knuth, Leslie Lamport, Timothy Van Zandt, Adobe Sys- tems, Larry Wall, the World Wide Web Consortium, and Maplesoft for those lan- guages. Thanks to Profs. Mark Skandera and Boris Pittel for explanations of the Neggers- Stanley conjecture. Prof. Herbert Wilf encouraged the work in Chapter IV. Prof. Jeff Lagarias found me a copy of [22]. Thanks to all my friends with Ph.D.’s who inspired me by example during my time in the wilderness. They include Paul Greiner, Rick Mohr, Thom Sterling, Jill Baker, Ray Bingham, Lucy Hadden, Will Brockman, Ming Ye, Xueqing Tan, Ivan Yourshaw, Ken Hodges, Su Fang Ng, John Remmers, Steven Lybrand, Wil Lybrand, Jan Wolter, and Larry Mohr. Thanks to Profs. Keith Riles and Roberto Merlin, who let me sit in on their physics classes in 1999. If they had been less encouraging, I might have gone back to pro- gramming for a living. Thanks to Prof. Al Taylor, who encouraged me to get back into the math program, and has acted as my unofficial protocol adviser throughout. Thanks to all the staff in the graduate math office at Michigan, especially Warren Noone, Tara McQueen, Christine Betz Bolang, Bert Ortiz, and Jennifer Wagner. Overcoming bureaucracy has always been a challenge for me, and it helped enor- mously to have advocates (and often surrogates) for dealing with the powers that be. The department also generously paid my tuition for the Winter 2006 semester. iii Thanks to Jayne London for fostering all the programs for graduate students over at Rackham, and for talking to me about options. Many thanks to my good friend and teaching partner Jason Howald, who was always encouraging and helpful when I was stuck, and who gave me the greatest gift one mathematician can give another: he listened to me talk about my problem, thought about it, and worked with me to find a different approach. Thanks to Profs. Persi Diaconis and Jason Fulman for conversations and tips on directions to take, and to Jim Reeds and Ed Gilbert for explaining the history to me. Thanks to Prof. Bruce Sagan, who helped me with mathematical as well as political advice on a number of occasions. Sergi Elizalde talked with me about EDπ(1) (Chapter IV, Theorem 4.3), and wrote a proof a few days after I did. Thanks to all my office mates: Kamran Kashef, Ken Keppen, Jared Maruskin, Jiarui Fei, Dave Constantine, and Janis Stipins, for putting up with my clutter (and my primitive origami). Thanks to Teresa Hunt for years of encouragement and for several pearls of wisdom and tricks for getting things done. Thanks to Nito for keeping me company. Thanks to my study partners of the past few years: Cornelia Yuen, Paul Greiner, Rob Houck, and Chris Bichler. iv Many thanks to my stepfather, Wil Lybrand, who gave me the invaluable advice to treat the Ph.D. as a union card, not a life’s work. Prof. Sergey Fomin taught me everything I know about Young diagrams and sym- metric functions, and he has always been generous with his time and his support. My advisor Divakar Viswanath, of course, was more involved with this work than anyone, and deserves the most credit for its success. We began work on the topic of card shuffling in 2002, when he gave me a copy of [3] to read as part of a class on Markov Chains. He suggested thinking about problems that the authors had not considered, and that was the source of the question about decks with repeated cards. Throughout the development of the ideas in this thesis he has usually been several steps ahead of me. My Mother, Drew Conger, has been unbelievably patient waiting for me to finish my degree. I learned not only patience but almost everything else I value from her. Carol Mohr has endured the most in service of this project. She has also given the most support, and for that I am forever grateful. v TABLE OF CONTENTS ACKNOWLEDGEMENTS .................................. ii LIST OF FIGURES ...................................... x LIST OF TABLES ....................................... xiii LIST OF APPENDICES ................................... xiv GLOSSARY ........................................... xv NOTATION ........................................... xx CHAPTER I. Introduction ....................................... 1 1.1 PreviousResults................................. 2 1.2 RepeatedCards .................................. 6 1.3 NewResults .................................... 8 1.3.1 Dealing is Equivalent to Fixing the Target Deck . ..... 8 1.3.2 Transition Probabilities and Descent Polynomials . ........ 9 1.3.3 Approximating Transition Probabilities . ...... 9 1.3.4 Bridge .................................. 11 1.3.5 Other Results from First-Order Approximations . ...... 12 1.3.6 MonteCarloSimulations. 13 1.3.7 Calculation of Descent Polynomials for Certain Decks ....... 13 vi 1.3.8 The Joint Distribution of des(π) and π(1) .............. 14 II. Preliminaries ....................................... 15 2.1 PermutationsandDecks . .. .. .. .. .. .. .. .. .. .. .. 15 2.2 RepeatedValues.................................. 16 2.3 MixingProblems.................................. 18 2.4 RiffleShuffling................................... 19 2.5 a-shuffles...................................... 22 2.6 InverseShufflesandRepeatedShuffles. .... 23 2.7 Counting Shuffles Which Produce a Permutation . ...... 26 2.8 DescentPolynomialsandShuffleSeries . ..... 29 2.9 DistancefromUniform ............................. 30 2.10 VariationDistance. ... 33 2.11 Dealing Cards is Equivalent to Fixing the Target Deck . .......... 36 2.12 HowGoodistheGSRModel? . 38 III. Probability Calculations for Some Simple Decks ................ 41 3.1 TheSimplestDeck ................................ 43 3.2 OneRedCardonTop............................... 44 3.3 OneRedCardontheBottom .......................... 45 3.4 AnyPositiontoTop................................ 47 3.5 AnyPositiontoBottom.. .. .. .. .. .. .. .. .. .. .. .. 49 3.6 AnyPositiontoAnyPosition . .. 50 3.7 OneRedCard,OneGreenCard . 52 3.8 Source deck RmBn ................................. 53 3.9 Target deck RmBn ................................. 56 3.10 Source Deck 1n1 2n2 ...mnm ........................... 57 vii 3.11 Target Deck 1n1 2n2 ...mnm ............................ 60 3.12 TargetDecksContainingBlocks . ..... 61 IV. The Joint Distribution of π(1) and des(π) in Sn ................. 65 4.1 IntroductionandMainResults. .... 65 4.2 BasicProperties ................................. 68 4.3 Recurrences .................................... 70 4.4 FormulasandMoments .............................. 71 4.5 Application Using Stein’s Method . ..... 77 4.6 GeneratingFunctions . .. .. .. .. .. .. .. .. .. .. .. .. .. 79 4.7 GeneralBehavior ................................. 85 4.8 Behavior if d n ................................. 88 4.9 IfBothEndsAreFixed.............................. 89 4.10Remarks ...................................... 92 V. Estimating Variation Distance ............................ 93 5.1 TheBasicQuestionsofCardShuffling. .... 93 5.2 Calculating Variation Distance Exactly . ........ 95 5.3 Expansion of the Transition Probability as a Power Series in a−1 ....... 96 5.4 Approximating the Transition Probability and Variation Distance for Large Shuffles....................................... 99 5.5 All-DistinctDecks............................... 102 5.6 Calculation of κ1 whentheSourceDeckisFixed. 105 5.6.1 TwoCardTypes ............................ 105 5.6.2 For Which Collections of Cards Can κ1 beZero? . 108 5.6.3 Guessing How Large κ1 CanBe....................111 5.6.4 Finding κ1 forGeneralSourceDecks . 115 5.7 Calculation of κ1 whentheTargetDeckisFixed . 118 viii 5.7.1 For Which Collections of Cards Can κ1 beZero? . 119 5.7.2 Finding κ1 forGeneralTargetDecks . 122 5.7.3 Euchre..................................123
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