Combinatorics of permutations [1 ed.] 1584884347, 9781584884347, 9781420035186

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DISCRETE MATHEMATICS AND ITS APPLICATIONS Series Editor

Kenneth H.Rosen, Ph.D. AT&T Laboratories Middletown, New Jersey Miklós Bóna, Combinatorics of Permutations Kun-Mao Chao and Bang Ye Wu, Spanning Trees and Optimization Problems Charalambos A.Charalambides, Enumerative Combinatorics Charles J.Colbourn and Jeffrey H.Dinitz, The CRC Handbook of Combinatorial Designs Steven Furino, Ying Miao, and Jianxing Yin, Frames and Resolvable Designs: Uses, Constructions, and Existence Randy Goldberg and Lance Riek, A Practical Handbook of Speech Coders Jacob E.Goodman and Joseph O’Rourke, Handbook of Discrete and Computational Geometry, Second Edition Jonathan Gross and Jay Yellen, Graph Theory and Its Applications Jonathan Gross and Jay Yellen, Handbook of Graph Theory Darrel R.Hankerson, Greg A.Harris, and Peter D.Johnson, Introduction to Information Theory and Data Compression, Second Edition Daryl D.Harms, Miroslav Kraetzl, Charles J.Colbourn, and John S.Devitt, Network Reliability: Experiments with a Symbolic Algebra Environment David M.Jackson and Terry I.Visentin, An Atlas of Smaller Maps in Orientable and Nonorientable Surfaces Richard E.Klima, Ernest Stitzinger, and Neil P.Sigmon, Abstract Algebra Applications with Maple Patrick Knupp and Kambiz Salari, Verification of Computer Codes in Computational Science and Engineering Donald L.Kreher and Douglas R.Stinson, Combinatorial Algorithms: Generation Enumeration and Search Charles C.Lindner and Christopher A.Rodgers, Design Theory Alfred J.Menezes, Paul C.van Oorschot, and Scott A.Vanstone, Handbook of Applied Cryptography Richard A.Mollin, Algebraic Number Theory Richard A.Mollin, Fundamental Number Theory with Applications

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Richard A.Mollin, An Introduction to Cryptography Richard A.Mollin, Quadratics Richard A.Mollin, RSA and Public-Key Cryptography Kenneth H.Rosen, Handbook of Discrete and Combinatorial Mathematics Douglas R.Shier and K.T.Wallenius, Applied Mathematical Modeling: A Multidisciplinary Approach Douglas R.Stinson, Cryptography: Theory and Practice, Second Edition Roberto Togneri and Christopher J.deSilva, Fundamentals of Information Theory and Coding Design Lawrence C.Washington, Elliptic Curves: Number Theory and Cryptography

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DISCRETE MATHEMATICS AND ITS APPLICATIONS Series Editor KENNETH H.ROSEN

Combinatorics of PERMUTATIONS Miklós Bóna

CHAPMAN & HALL/CRC A CRC Press Company Boca Raton London New York Washington, D.C.

© 2004 by Chapman & Hall/CRC

Library of Congress Cataloging-in-Publication Data Bóna, Miklós. Combinatorics of permutations/Miklós, Bóna. p. cm.—(Discrete mathematics and its applications) Includes bibliographical references and index. ISBN 1-58488-434-7 (alk. paper) 1. Permutations. I. Title. II. Series. QA165.B66 2004 511'.64—dc22

2004045868

This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage or retrieval system, without prior permission in writing from the publisher. The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for creating new works, or for resale. Specific permission must be obtained in writing from CRC Press LLC for such copying. Direct all inquiries to CRC Press LLC, 2000 N.W. Corporate Blvd., Boca Raton, Florida 33431. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation, without intent to infringe.

Visit the CRC Press Web site at www.crcpress.com © 2004 by Chapman & Hall/CRC No claim to original U.S. Government works International Standard Book Number 1-58488-434-7 Library of Congress Card Number 2004045868 Printed in the United States of America 1 2 3 4 5 6 7 8 9 0 Printed on acid-free paper

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Foreword

FOREWORD Permutations have a remarkably rich combinatorial structure. Part of the reason for this is that a permutation of a finite set can be represented in many equivalent ways, including as a word (sequence), a function, a collection of disjoint cycles, a matrix, etc. Each of these representations suggests a host of natural invariants (or “statistics”), operations, transformations, structures, etc., that can be applied to or placed on permutations. The fundamental statistics, operations, and structures on permutations include descent set (with numerous specializations), excedance set, cycle type, records, subsequences, composition (product), partial orders, simplicial complexes, probability distributions, etc. How is the newcomer to this subject able to make sense of and sort out these bewildering possibilities? Until now it was necessary to consult a myriad of sources, from textbooks to journal articles, in order to grasp the whole picture. Now, however, Miklós Bóna has provided us with a comprehensive, engaging, and eminently readable introduction to all aspects of the combinatorics of permutations. The chapter on pattern avoidance is especially timely and gives the first systematic treatment of this fascinating and active area of research. This book can be utilized at a variety of levels, from random samplings of the treasures therein to a comprehensive attempt to master all the material and solve all the exercises. In whatever direction the reader’s tastes lead, a thorough enjoyment and appreciation of a beautiful area of combinatorics is certain to ensue. Richard Stanley Cambridge, Massachusetts January 14, 2004

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Preface

A few years ago, I was given the opportunity to teach a graduate Combinatorics class on a special topic of my choice. I wanted the class to focus on the Combinatorics of Permutations. However, I instantly realized that while there were several excellent books that discussed some aspects of the subject, there was no single book that would have contained all, or even most, areas that I wanted to cover. Many areas were not covered in any book, which was easy to understand as the subject is developing at a breathtaking pace, producing new results faster than textbooks are published. Classic results, while certainly explained in various textbooks of very high quality, seemed to be scattered in numerous sources. This was again no surprise; indeed, permutations are omnipresent in modern combinatorics, and there are quite a few ways to look at them. We can consider permutations as linear orders, we can consider them as elements of the symmetric group, we can model them by matrices, or by graphs. We can enumerate them according to countless interesting statistics, we can decompose them in many ways, and we can bijectively associate them to other structures. One common feature of these activities is that they all involve factual knowledge, new ideas, and serious fun. Another common feature is that they all evolve around permutations, and quite often, the remote-looking areas are connected by surprising results. Briefly, they do belong to one book, and I am very glad that now you are reading such a book. *** As I have mentioned, there are several excellent books that discuss various aspects of permutations. Therefore, in this book, I cover these aspects less deeply than the areas that had previously not been contained in any book. Chapter 1 is about descents and runs of permutations. While Eulerian numbers have been given plenty of attention during the last 200 years, most of the research was devoted to analytic concepts. Nothing shows this better than the fact that I was unable to find published proofs of two fundamental results of the area using purely combinatorial methods. Therefore, in this Chapter, I concentrated on purely combinatorial tools dealing with these issues. By and large, the same is true for Chapter 2. Chapter 3 is devoted to permutations as products of cycles, which is probably the most-studied of all areas covered in this book. Therefore, there were many classic results we had to include there for the sake of

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completeness, nevertheless we still managed to squeeze in less well-known topics, such as applications of Darroch’s theorem, or transpositions and trees. The area of pattern avoidance is a young one, and has not been given significant space in textbooks before. Therefore, we devoted two full chapters to it. Chapter 4 walks the reader through the quest for the solution of the Stanley-Wilf conjecture, ending with the recent spectacular proof of Marcus and Tardos for this 23-yearold problem. Chapter 5 discusses aspects of pattern avoidance other than upper bounds or exact formulae. Chapter 6 looks at random permutations and Standard Young Tableaux, starting with two classic and difficult proofs of Greene, Nijenhaus and Wilf. Standard techniques for handling permutation statistics are presented. A relatively new concept, that of min-wise independent families of permutations, is discussed in the Exercises. Chapter 7, Algebraic Combinatorics of Permutations, is the one in which we had to be most selective. Each of the three sections of that chapter covers an area that is sufficiently rich to be the subject of an entire book. Our goal with that chapter is simply to raise interest in these topics and prepare the reader for the more detailed literature that is available in those areas. Finally, Chapter 8 is about combinatorial sorting algorithms, many of which are quite recent. This is the first time many of these algorithms (or at least, most aspects of them) are discussed in a textbook, so we treated them in depth. Besides the Exercises, each Chapter ends with a selection of Problems Plus. These are typically more difficult than the exercises, and are meant to raise interest in some questions for further research, and to serve as reference material of what is known. Some of the Problems Plus are not classified as such because of their level of difficulty, but because they are less tightly connected to the topic at hand. A solution manual for the even-numbered Exercises is available for instructors teaching a class using this book, and can be obtained from the publisher.

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Acknowledgments

This book grew out of various graduate combinatorics courses that I taught at the University of Florida. I am indebted to the authors of the books I used in those courses, for shaping my vision, and for teaching me facts and techniques. This books are “The Art of Computer Programming” by D.E.Knuth, “Enumerative Combinatorics” by Richard Stanley, “The Probabilistic Method” by Noga Alon and Joel Spencer, “The Symmetric Group” by Bruce Sagan, and “Enumerative Combinatorics” by Charalambos Charalambides. Needless to say, I am grateful to all the researchers whose results made a textbook devoted exclusively to the combinatorics of permutations possible. I am sure that new discoveries will follow. I am thankful to my former research advisor Richard Stanley for having introduced me into this fascinating field, and to Herb Wilf and Doron Zeilberger, who kept asking intriguing questions attracting scores of young mathematicians like myself to the subject. Some of the presented material was part of my own research, sometimes in collaboration. I would like to say thanks to my co-authors, Richard Ehrenborg, Andrew MacLennan, Bruce Sagan, Rodica Simion, Daniel Spielman, Vincent Vatter, and Dennis White. I also owe thanks to Michael Atkinson, who introduced me into the history of stack sorting algorithms. I am deeply indebted to Aaron Robertson for an exceptionally thorough and knowledgeable reading of my first draft. I am also deeply appreciative for manuscript reading by my colleague Andrew Vince, and by Rebecca Smith. A significant part of the book was written during the summer of 2003. In the first half of that summer, I enjoyed the stimulating professional environment at LABRI, at the University of Bordeaux I, in Bordeaux, France. The hospitality of colleagues Olivier Guibert and Sylvain Pelat-Alloin made it easy for me to keep writing during my one-month visit. In the second half of the summer, I enjoyed the hospitality of my parents, Miklós and Katalin Bóna, at the Lake Balaton in Hungary. My gratitude is extended to Joseph Sciacca, who prepared the second cover page for a book of mine within two years. Last, but not least, I must be thankful to my wife Linda, my first reader and critic, who tolerated surprisingly well that I wrote a book again. I will not forget how much she helped me, and neither will she.

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Dedication

To Linda, Mikike, Benjamin, and my future children. To the Mathematicians whose relentless and brilliant efforts throughout the centuries unearthed the gems that we call Combinatorics of Permutations. The Tribute of the Current to the Source. Robert Frost, West Running Brook

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Contents

No Way Around It. Introduction.

1

1 In One Line And Close. Permutations as Linear Orders. Runs. 1.1 Descents 1.1.1 The definition of descents 1.1.2 Eulerian numbers 1.1.3 Stirling numbers and Eulerian numbers 1.1.4 Generating functions and Eulerian numbers 1.1.5 The sequence of Eulerian numbers 1.2 Alternating runs Exercises Problems Plus Solutions to Problems Plus

3 3 3 4 11 14 16 24 31 36 38

2 In One Line And Anywhere. Permutations as Linear Orders. Inversions. 2.1 Inversions 2.1.1 The generating function of permutations by inversions 2.1.2 Major index 2.1.3 An Application: Determinants and Graphs 2.2 Inversions in Permutations of Multisets 2.2.1 Inversions and Gaussian Coefficients 2.2.2 Major Index and Permutations of Multisets Exercises Problems Plus Solutions to Problems Plus

43 43 43 52 55 57 60 61 64 67 69

3 In Many Circles. Permutations as Products of Cycles. 3.1 Decomposing a permutation into cycles 3.1.1 An Application: Sign and Determinants 3.1.2 An Application: Geometric transformations 3.2 Type and Stirling numbers 3.2.1 The type of a permutation 3.2.2 An Application: Conjugate permutations 3.2.3 An Application: Trees and Transpositions

73 73 75 78 79 79 80 81

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xiv

Combinatorics of Permutations

3.2.4 Permutations with a given number of cycles 3.2.5 Generating functions for Stirling numbers 3.2.6 An Application: Real Zeros and Probability 3.3 Cycle Decomposition versus Linear Order 3.3.1 The Transition Lemma 3.3.2 Applications of the Transition Lemma 3.4 Permutations with restricted cycle structure 3.4.1 The exponential formula 3.4.2 The cycle index and its applications Exercises Problems Plus Solutions to Problems Plus

85 92 95 96 96 98 100 100 110 115 120 123

4 In Any Way But This. Pattern Avoidance. The Basics. 4.1 The notion of Pattern avoidance 4.2 Patterns of length three 4.3 Monotone Patterns 4.4 Patterns of length four 4.4.1 The Pattern 1324 4.4.2 The Pattern 1342 4.4.3 The Pattern 1234 4.5 The Proof of The Stanley-Wilf Conjecture 4.5.1 The Füredi-Hajnal conjecture 4.5.2 Avoiding Matrices vs. Avoiding Permutations 4.5.3 The Proof of the Füredi-Hajnal conjecture Exercises Problems Plus Solutions to Problems Plus

129 129 130 133 135 137 144 158 159 159 160 161 164 168 170

5 In This Way, But Nicely. Pattern Avoidance. Followup. 5.1 Polynomial Recursions 5.1.1 Polynomially Recursive Functions 5.1.2 Closed Classes of Permutations 5.1.3 Algebraic and Rational Power Series 5.1.4 The P-recursiveness of Sn,r(132) 5.2 Containing a pattern many times 5.2.1 Packing Densities 5.2.2 Layered Patterns 5.3 Containing a pattern a given number of times 5.3.1 A Construction With a Given Number of Copies 5.3.2 The sequence {kn}n≥0 Exercises Problems Plus Solutions to Problems Plus

175 175 175 176 178 182 191 191 193 198 199 201 205 207 208

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Table of Contents

xv

6 Mean and Insensitive. Random Permutations. 6.1 The Probabilistic Viewpoint 6.1.1 Standard Young Tableaux 6.2 Expectation 6.2.1 Linearity of Expectation 6.3 Variance and Standard Deviation 6.4 An Application: Longest Increasing Subsequences Exercises Problems Plus Solutions to Problems Plus

213 213 214 229 231 233 237 238 242 243

7 Permutations vs. Everything Else. Algebraic Combinatorics of Permutations. 7.1 The Robinson-Schensted-Knuth correspondence 7.2 Posets of permutations 7.2.1 Posets on Sn 7.2.2 Posets on Pattern Avoiding Permutations 7.2.3 An Infinite Poset of Permutations 7.3 Simplicial Complexes of permutations 7.3.1 A Simplicial Complex of Restricted Permutations 7.3.2 A Simplicial Complex of All n-Permutations Exercises Problems Plus Solutions to Problems Plus

247 247 257 257 265 267 269 269 271 272 276 278

8 Get Them All. Algorithms and Permutations. 8.1 Generating Permutations 8.1.1 Generating All n-permutations 8.1.2 Generating Restricted Permutations 8.2 Stack Sorting Permutations 8.2.1 2-Stack Sortable Permutations 8.2.2 t-Stack Sortable Permutations 8.2.3 Unimodality 8.3 Variations Of Stack Sorting Exercises Problems Plus Solutions to Problems Plus

283 283 283 284 287 289 291 297 300 307 311 313

Do Not Look Just Yet. Solutions to Odd-numbered Exercises. Solutions for Chapter 1 Solutions for Chapter 2 Solutions for Chapter 3 Solutions for Chapter 4 Solutions for Chapter 5 Solutions for Chapter 6

319 319 326 330 339 347 351

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xvi Solutions for Chapter 7 Solutions for Chapter 8

Combinatorics of Permutations 355 358

References

363

List of Frequently Used Notations

377

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No Way Around It. Introduction.

This book is devoted to the study of permutations. While the overwhelming majority of readers already know what they are, we are going to define them for the sake of completeness. Note that this is by no means the only definition possible. DEFINITION 0.1 A linear ordering of the elements of the set [n]= {1, 2, 3, ···, n} is called a permutation, or, if we want to stress the fact that it consists of n entries, an npermutation. In other words, a permutation lists all elements of [n] so that each element is listed exactly once. Example 0.2 If n=3, then the n-permutations are 123, 132, 213, 231, 312, 321. There is nothing magic about the set [n], other sets having n elements would be just as good for our purposes, but working with [n] will simplify our discussion. In Chapter 2, we will extend the definition of permutations onto multisets, and in Chapter 3, we will work with an alternative concept of looking at permutations. For now, we will denote an n-permutations by p=p1p2···pn, with pi being the ith entry in the linear order given by p. The following simple statement is probably the best-known fact about permutations. PROPOSITION 0.3 The number of n-permutations is n!. PROOF When building an n-permutation p=p1p2···pn, we can choose n entries to play the role of p1, then n-1 entries for the role of p2, and so on. I promise the rest of the book will be less straightforward.

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1 In One Line And Close. Permutations as Linear Orders. Runs.

1.1 Descents The “most orderly” of all n-permutations is obviously the increasing permutation 123···n. All other permutations have at least some “disorder” in them, for instance, it happens that an entry is immediately followed by a smaller entry in them. This simple phenomenon is at the center of our attention in this Section. 1.1.1 The definition of descents DEFINITION 1.1 Let p=p1p2···pn be a permutation. We say that i is a descent of p if pi>pi+1. Similarly, we say that i is an ascent of p if pi