UC San Diego UC San Diego Electronic Theses and Dissertations Title Extensions of the Reciprocity Method in Consecutive Pattern Avoidance in Permutations Permalink https://escholarship.org/uc/item/26c5p01d Author Bach, Quang Tran Publication Date 2017 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California UNIVERSITY OF CALIFORNIA, SAN DIEGO Extensions of the Reciprocity Method in Consecutive Pattern Avoidance in Permutations A Dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Mathematics by Quang Tran Bach Committee in charge: Professor Jeffrey Remmel, Chair Professor Ronald Graham Professor Ramamohan Paturi Professor Brendon Rhoades Professor Jacques Verstraete 2017 Copyright Quang Tran Bach, 2017 All rights reserved. The Dissertation of Quang Tran Bach is approved, and it is acceptable in quality and form for publication on microfilm and electronically: Chair University of California, San Diego 2017 iii DEDICATION To my little brother, my trusty friend and loyal companion. iv EPIGRAPH Once you eliminate the impossible, whatever remains, no matter how improbable, must be the truth. —Sir Arthur Conan Doyle, Sherlock Holmes v TABLE OF CONTENTS Signature Page................................... iii Dedication...................................... iv Epigraph......................................v Table of Contents.................................. vi List of Figures................................... viii List of Tables....................................x Acknowledgements................................. xi Vita......................................... xv Abstract of the Dissertation............................ xvi Chapter 1 Introduction.............................1 1.1 A quick history of permutation patterns..........1 1.2 Symmetric functions and brick tabloids...........7 1.3 The main goal of this thesis................. 16 Chapter 2 The reciprocity method....................... 22 2.1 Pattern matching in the cycle structure of permutations. 23 2.2 The reciprocity method.................... 28 2.3 Results of the reciprocity method.............. 35 2.3.1 The case Γ = Γk1;k2 .................. 35 2.3.2 Adding an identity permutation to Γk1;k2 ...... 44 2.3.3 The cases f1324; 123g and f1324 : : : p; 123 : : : p − 1g for p ≥ 5 ........................ 56 Chapter 3 The case of multiple descents................... 74 3.1 A new involution....................... 75 3.2 Results of the new involution................. 82 3.2.1 The case Γ = f14253; 15243g ............. 84 3.2.2 The case Γ = f142536g ................ 91 3.2.3 The proof of Theorem 12.............. 106 3.2.4 The remaining cases of τ = 152634; τ = 152436; τ = 162435; and τ = 142635 ............... 115 vi Chapter 4 Refinements of the c-Wilf equivalent relation........... 131 4.1 The proof of Theorem 18................... 138 4.2 The proof of Theorem 19................... 151 Chapter 5 Generating Function for Initial and Final Descents........ 156 5.1 The proof of Theorem 22................... 158 5.2 Generating function for the number of initial and final descents165 Bibliography.................................... 175 vii LIST OF FIGURES Figure 1.1: The six (12; 22)-brick tabloids of shape (6)..............7 Figure 1.2: The weights of the six (12; 22)-brick tabloids of shape (6)..... 11 Figure 1.3: The (12; 22)-brick tabloids of shape (6) under multiple weights.. 13 Figure 2.1: The construction of a filled-labeled-brick tabloid.......... 30 Figure 2.2: IΓ(O) for O in Figure 2.1....................... 32 Figure 2.3: Patterns for two consecutive brick of size 2 in a fixed point of IΓ. 62 Figure 2.4: The bijection φ............................ 63 Figure 2.5: The bijection θn;k........................... 64 −1 Figure 2.6: The bijection θn;k........................... 65 Figure 2.7: The flip of Dyck path......................... 66 Figure 2.8: The bijection βn;k........................... 67 −1 Figure 2.9: The bijection βn;k........................... 68 Figure 2.10: A fixed point with Γp-matches starting at ci for i = 1; : : : ; m − 1. 71 Figure 3.1: An example of the involution JΓ................... 77 Figure 3.2: A fixed point of Jf15342g........................ 82 Figure 3.3: The possible choice for d in Subcase 2a............... 88 Figure 3.4: Subcase 2b............................... 90 Figure 3.5: A 142536-match as a 2-line array................... 93 Figure 3.6: Fixed points that start with series of τ-matches........... 95 Figure 3.7: Fixed points that start with series of τ-matches in Case 2.1.... 96 ¯ Figure 3.8: The Hasse diagram of G6k+4 for k = 0; 1; 2............. 98 Figure 3.9: The Hasse diagram of Dn....................... 98 ¯ Figure 3.10: Partitioning the Hasse Diagram of G6k+4.............. 99 Figure 3.11: The matrix M7............................. 100 Figure 3.12: Fixed points that start with series of τ-matches in Case 2.2.... 100 Figure 3.13: i starts brick b2k+3........................... 102 ¯ Figure 3.14: The Hasse diagram of G6k+2 for k = 0; 1; 2............. 104 ¯ Figure 3.15: Partitioning the Hasse Diagram of G6k+2.............. 105 Figure 3.16: The matrix P7............................. 106 Figure 3.17: The Hasse diagram associated with τa................ 106 Figure 3.18: The Hasse diagram of overlapping τa-mathces............ 107 Figure 3.19: Subcases (2.A) and (2.B)....................... 110 Figure 3.20: The ordering of fσ2; σ4; : : : ; σ4kg for k = 2 (left) and for general k (right).................................. 118 Figure 3.21: The ordering of fσ1; σ2; : : : ; σ10g in S2................ 120 Figure 3.22: The ordering of fσ1; σ2; : : : ; σ4k+2g in Sk (top) and its simplified structure (bottom)........................... 122 Figure 3.23: The generalized diagram for L(k; n)................. 122 Figure 3.24: The ordering of the first 4k + 2 cells of O for τ = 142635..... 127 viii Figure 3.25: The Hasse diagram for L(a1; : : : ; an+1; b1; : : : ; bn; c1; : : : ; cn): ... 128 Figure 4.1: The construction of a filled, labeled brick tabloid.......... 141 Figure 5.1: An example of the object created from equation (5.1)....... 160 Figure 5.2: The image of the filled, labeled brick tabloid from Figure 5.1... 162 Figure 5.3: A fixed point of the involution I................... 163 Figure 5.4: A “regular" fixed point of the involution I0............. 170 ix LIST OF TABLES Table 2.1: The polynomials UΓ2;2;n(−y) for Γ2;2 = f1324; 1423g ........ 42 Table 2.2: The polynomials UΓ6;4;n(−y) ..................... 44 Table 2.3: The polynomials UΓ2;2;2;2k(−y) for Γ2;2;2 = f1324; 1423; 123g .... 49 Table 2.4: The polynomials UΓ2;2;2;2k+1(−y) for Γ2;2;2 = f1324; 1423; 123g ... 49 Table 2.5: The polynomials UΓ2;2;3;3k(−y) for Γ2;2;3 = f1324; 1423; 1234g ... 51 Table 2.6: The polynomials UΓ2;2;3;3k+1(−y) for Γ2;2;3 = f1324; 1423; 1234g .. 52 Table 2.7: The polynomials UΓ2;2;3;3k+2(−y) for Γ2;2;3 = f1324; 1423; 1234g .. 52 Table 2.8: The polynomials UΓ;n(−y) for Γ = f1324; 123g ........... 60 Table 3.1: The polynomials UΓ;n(−y) for Γ = f14253; 15243g ......... 90 Table 3.2: The polynomials MNΓ;n(x; y) for Γ = f14253; 15243g ....... 91 Table 3.3: The polynomials Uτ;n(y) for τ = 142536: .............. 107 Table 3.4: The first eight values of Sk...................... 123 Table 3.5: The first eight values of Lk...................... 129 Table 4.1: The c-Wilf equivalent classes of length 5............... 150 x ACKNOWLEDGEMENTS First and foremost, I would like to thank my family for their unconditional love and support. I have been extremely fortunate in my life to have such an amazing and unique family. My parents have sacrificed a lot in order for me to have a better life and education. To this day, I still remember my Dad B¤ch Công Sơn making trips of 60 kilometers one way between home and work almost every day of the week. Dad has been spending more than 30 years of his life riding on an old scooter amidst the wind, sand, and dust of the polluted rural areas in Vietnam and under the harsh sunlight and downpour rain of a tropical country just to be with the family and to take me to school in the morning. My Mom Tr¦n Thị Thi·u and my Aunt Tr¦n Thị Thoa have been working non-stop from 5:00 a.m. to midnight everyday in order to provide me a perfect childhood with the best education. To my family in the United States, I would like to thank my Uncle Tr¦n Gia Tu§n for his tremendous amount of help and counsel with my everyday problems as I am trying to adapt and integrate to the new life in a foreign country. In fact, Uncle Tu§n has been a father figure for me in my father's absence. I would like to thank my Grandpa Tr¦n Gia T¡, my Aunts Tr¦n Tina, Tôn Nú Lan Hương, and Tr¦n Kim Trúc for treating me as their own son, being constantly concerned about my well-being, and for putting food on the table and a roof over my head over the past ten years. I owe a tremendous amount of gratitude to all the members of my family as they have played an important role in the development of my identity and shaping the individual that I am today. I would like to thank my thesis advisor, Professor Jeffrey Remmel, who has supported me throughout my graduate studies with his patience, knowledge, and wisdom. It has been an utmost honor to study under the guidance of a brilliant xi mathematician and dedicated educator like Professor Remmel. He has ignited in me the joy, passion, and enthusiasm for the study of enumerative combinatorics and permutation patterns. Professor Remmel has also contributed greatly to my rewarding experience at the University of California, San Diego (UCSD) by engaging me with numerous intellectual ideas and research projects, funding my trips to various meetings and conferences, referring me to many employment opportunities to develop my curriculum vitae, providing tips and feedbacks for my teaching skills, and always demanding a high quality of work in all of our papers.