Statistics on Pattern-avoiding Permutations by Sergi Elizalde B.S., Technical University of Catalonia (UPC), 2000 Submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2004 c Sergi Elizalde, MMIV. All rights reserved. The author hereby grants to MIT permission to reproduce and distribute publicly paper and electronic copies of this thesis document in whole or in part. Author . Department of Mathematics April 22, 2004 Certified by. Richard P. Stanley Norman Levinson Professor of Applied Mathematics Thesis Supervisor Accepted by . Pavel Etingof Chairman, Department Committee on Graduate Students 2 Statistics on Pattern-avoiding Permutations by Sergi Elizalde Submitted to the Department of Mathematics on April 22, 2004, in partial fulfillment of the requirements for the degree of Doctor of Philosophy Abstract This thesis concerns the enumeration of pattern-avoiding permutations with respect to certain statistics. Our first result is that the joint distribution of the pair of statistics `number of fixed points' and `number of excedances' is the same in 321-avoiding as in 132-avoiding permuta- tions. This generalizes a recent result of Robertson, Saracino and Zeilberger, for which we also give another, more direct proof. The key ideas are to introduce a new class of statis- tics on Dyck paths, based on what we call a tunnel, and to use a new technique involving diagonals of non-rational generating functions. Next we present a new statistic-preserving family of bijections from the set of Dyck paths to itself. They map statistics that appear in the study of pattern-avoiding permutations into classical statistics on Dyck paths, whose distribution is easy to obtain. In particular, this gives a simple bijective proof of the equidistribution of fixed points in the above two sets of restricted permutations. Then we introduce a bijection between 321- and 132-avoiding permutations that pre- serves the number of fixed points and the number of excedances. A part of our bijection is based on the Robinson-Schensted-Knuth correspondence. We also show that our bijection preserves additional parameters. Next, motivated by these results, we study the distribution of fixed points and ex- cedances in permutations avoiding subsets of patterns of length 3. We solve all the cases of simultaneous avoidance of more than one pattern, giving generating functions which enu- merate them. Some cases are generalized to patterns of arbitrary length. For avoidance of one single pattern we give partial results. We also describe the distribution of these statis- tics in involutions avoiding any subset of patterns of length 3. The main technique consists in using bijections between pattern-avoiding permutations and certain kinds of Dyck paths, in such a way that the statistics in permutations that we consider correspond to statistics on Dyck paths which are easier to enumerate. Finally, we study another kind of restricted permutations, counted by the Motzkin num- bers. By constructing a bijection into Motzkin paths, we enumerate them with respect to some parameters, including the length of the longest increasing and decreasing subsequences and the number of ascents. Thesis Supervisor: Richard P. Stanley Title: Norman Levinson Professor of Applied Mathematics 3 4 Acknowledgments There are many people who have made this thesis possible. First I would like to thank my advisor, Richard Stanley, for his guidance and advice, and for always pointing me in the right direction. I feel very lucky to have been one of his students. I admire not only his tremendous knowledge, but also his outstanding humbleness. I will try to follow his model, wishing that one day I can be such a good advisor. The stimulating environment at MIT has allowed me to learn from many people, both professors and students. In particular, I want to thank Igor Pak for mathematical dis- cussions and collaboration, and for his liveliness and sense of humor. It has also been a pleasure to collaborate with Emeric Deutsch and Toufik Mansour. Other mathematicians who have offered valuable suggestions are Sara Billey, Miklos B´ona, Alex Burstein, Richard Ehrenborg, Ira Gessel, Olivier Guibert, David Jackson, Sergey Kitaev, Danny Kleitman, Rom Pinchasi, Alex Postnikov, Astrid Reifegerste and Douglas Rogers. Outside of MIT, I am very grateful to Marc Noy for previous collaboration, and for following my work and giving me priceless advice in my frequent visits to Barcelona. He awakened my interest for combinatorics, soon after Josep Gran´e and Sebasti`a Xamb´o had introduced me to the wonders of mathematics. These four years at MIT have been one of the best periods of my life, both scientifically and at a personal level. I owe this to the exceptional people that I have met in Boston. I am indebted to Jan and Radoˇs for being such supportive friends, with whom one can talk about everything. From Federico I learned how to enjoy being a graduate student. I owe to Mark having given me the encouragement I needed to come to MIT, as well as other good advice. Peter has been much more than a \cofactor". Fumei has made the office more lively. Anna has helped me keep things in perspective, and has made me feel closer to UPC. My stay at MIT has been a great experience also thanks to Etienne, Carly, Bridget, Lauren, Cilanne, Thomas, Ed and Jason, and outside the math department, to Pinar, Hazhir, Parisa, Charlotte, Samantha, Kalina, Martin, V´ıctor, Rafal, Felipe, Cornelius, Jos´e Manuel, Mika, Nuria,´ Ram´on, Marta, Paulina, Juan, Carina, Anya, Han, Alessandra, Karolina, Stephanie, and the ones that I shamefully forgot. Finalment, l'agra¨ıment m´es especial ´es pels meus pares Emili i Maria Carme, sense els quals tot aix`o no hauria estat possible. A ells els dec l’educaci´o que m'han donat, haver-me format com a persona, i especialment l'amor que han demostrat i el seu suport incondicional en tot moment. Tamb´e ´es un orgull tenir un germ`a com l'Aleix, sabent que puc comptar amb ell sempre que el necessiti. Per acabar, vull donar gr`acies als meus amics de la carrera: Diego, Sergi, Javi, Marta, Agus, Teresa, Ariadna, Carles, Ana, Fernando, V´ıctor, Carme, Josep Joan, Montse, Toni, Edgar, Jordi, Maite, Esther, Desi, Elena i tots els altres, per fer-me sentir com si el temps no pass´es cada vegada que vaig a Barcelona. 5 6 Contents Introduction 13 1 Definitions and preliminaries 15 1.1 Permutations . 15 1.1.1 Pattern avoidance . 15 1.1.2 Permutation statistics . 15 1.1.3 Trivial operations . 16 1.2 Dyck paths . 17 1.2.1 Standard statistics . 18 1.2.2 Tunnels . 18 1.3 Combinatorial classes and generating functions . 21 1.3.1 The Lagrange inversion formula . 21 1.3.2 Chebyshev polynomials . 22 1.4 Patterns of length 3 . 22 1.4.1 Equidistribution of fixed points . 22 2 Equidistribution of fixed points and excedances 25 2.1 The bijection x . 25 2.2 The bijection ' . 28 2.2.1 Enumeration of centered tunnels . 30 2.2.2 Enumeration of shifted tunnels . 31 2.3 Enumeration of right tunnels: the method of the diagonal of a power series 32 2.4 Some other bijections between (321) and . 38 Sn Dn 3 A simple and unusual bijection for Dyck paths 41 3.1 The bijection Φ . 41 3.2 Properties of Φ . 44 3.3 Generalizations . 47 3.4 Connection to pattern-avoiding permutations . 50 3.5 Some new interpretations of the Catalan numbers . 54 4 A direct bijection preserving fixed points and excedances 57 4.1 A composition of bijections into Dyck paths . 57 4.1.1 The bijection Ψ . 58 4.2 Properties of Ψ . 59 4.3 Properties of the matching algorithm . 61 4.4 Further applications . 63 7 5 Avoidance of subsets of patterns of length 3 65 5.1 More properties of ' . 65 5.2 Single restrictions . 67 5.2.1 a) 123 . 67 5.2.2 b) 132 213 321 . 70 ≈ ≈ 5.2.3 c, c') 231 312 . 73 ∼ 5.3 Double restrictions . 77 5.3.1 a) 123; 132 123; 213 . 77 f g ≈ f g 5.3.2 b, b') 231; 321 312; 321 . 81 f g ∼ f g 5.3.3 c) 132; 213 . 82 f g 5.3.4 d) 231; 312 . 84 f g 5.3.5 e, e') 132; 231 213; 231 132; 312 213; 312 . 84 f g ≈ f g ∼ f g ≈ f g 5.3.6 f) 132; 321 213; 321 . 85 f g ≈ f g 5.3.7 g, g') 123; 231 123; 312 . 86 f g ∼ f g 5.3.8 h) 123; 321 . 88 f g 5.4 Triple restrictions . 88 5.5 Pattern-avoiding involutions . 94 5.5.1 Single restrictions . 95 5.5.2 Multiple restrictions . 96 5.6 Expected number of fixed points . 97 5.6.1 a) 123 . 98 5.6.2 b) 132 213 321 . 98 ≈ ≈ 5.6.3 c, c') 231 312 . 99 ∼ 5.6.4 Other cases . 99 5.7 Final remarks and possible extensions . 100 5.7.1 Cycle structure . 100 6 Motzkin permutations 103 6.1 Preliminaries . 103 6.1.1 Generalized patterns . 103 6.1.2 Motzkin paths . 104 6.2 The bijection Υ . 104 6.2.1 Definition of Υ . 104 6.2.2 Statistics on Mn . 105 6.3 Fixed points in the reversal of Motzkin permutations . 107 8 List of Figures 1-1 The array of π = 63528174. 16 1-2 One centered and four right tunnels. 19 1-3 Five centered multitunnels, two of which are centered tunnels.