FIBONACCI AND CATALAN NUMBERS FIBONACCI AND CATALAN NUMBERS AN INTRODUCTION Ralph P. Grimaldi Rose-Hulman Institute of Technology Copyright © 2012 by John Wiley & Sons, Inc. All rights reserved Published by John Wiley & Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permission. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic formats. For more information about Wiley products, visit our web site at www.wiley.com. Library of Congress Cataloging-in-Publication Data: Grimaldi, Ralph P. Fibonacci and catalan numbers : an introduction / Ralph P. Grimaldi. p. cm. Includes bibliographical references and index. ISBN 978-0-470-63157-7 1. Fibonacci numbers. 2. Recurrent sequences (Mathematics) 3. Catalan numbers (Mathematics) 4. Combinatorial analysis. I. Title. QA241.G725 2012 512.7’2–dc23 2011043338 Printed in the United States of America 10987654321 Dedicated to the Memory of Josephine and Joseph and Mildred and John and Madge CONTENTS PREFACE xi PART ONE THE FIBONACCI NUMBERS 1. Historical Background 3 2. The Problem of the Rabbits 5 3. The Recursive Definition 7 4. Properties of the Fibonacci Numbers 8 5. Some Introductory Examples 13 6. Compositions and Palindromes 23 7. Tilings: Divisibility Properties of the Fibonacci Numbers 33 8. Chess Pieces on Chessboards 40 9. Optics, Botany, and the Fibonacci Numbers 46 10. Solving Linear Recurrence Relations: The Binet Form for Fn 51 11. More on α and β: Applications in Trigonometry, Physics, Continued Fractions, Probability, the Associative Law, and Computer Science 65 12. Examples from Graph Theory: An Introduction to the Lucas Numbers 79 13. The Lucas Numbers: Further Properties and Examples 100 14. Matrices, The Inverse Tangent Function, and an Infinite Sum 113 15. The gcd Property for the Fibonacci Numbers 121 vii viii CONTENTS 16. Alternate Fibonacci Numbers 126 17. One Final Example? 140 PART TWO THE CATALAN NUMBERS 18. Historical Background 147 19. A First Example: A Formula for the Catalan Numbers 150 20. Some Further Initial Examples 159 21. Dyck Paths, Peaks, and Valleys 169 22. Young Tableaux, Compositions, and Vertices and Arcs 183 23. Triangulating the Interior of a Convex Polygon 192 24. Some Examples from Graph Theory 195 25. Partial Orders, Total Orders, and Topological Sorting 205 26. Sequences and a Generating Tree 211 27. Maximal Cliques, a Computer Science Example, and the Tennis Ball Problem 219 28. The Catalan Numbers at Sporting Events 226 29. A Recurrence Relation for the Catalan Numbers 231 30. Triangulating the Interior of a Convex Polygon for the Second Time 236 31. Rooted Ordered Binary Trees, Pattern Avoidance, and Data Structures 238 32. Staircases, Arrangements of Coins, The Handshaking Problem, and Noncrossing Partitions 250 33. The Narayana Numbers 268 34. Related Number Sequences: The Motzkin Numbers, The Fine Numbers, and The Schroder¨ Numbers 282 CONTENTS ix 35. Generalized Catalan Numbers 290 36. One Final Example? 296 Solutions for the Odd-Numbered Exercises 301 Index 355 PREFACE In January of 1992, I presented a minicourse at the joint national mathematics meetings held that year in Baltimore, Maryland. The minicourse had been approved by a com- mittee of the Mathematical Association of America—the mission of that committee being the evaluation of proposed minicourses. In this case, the minicourse was espe- cially promoted by Professor Fred Hoffman of Florida Atlantic University. Presented in two two-hour sessions, the first session of the minicourse touched upon examples, properties, and applications of the sequence of Fibonacci numbers. The second part investigated comparable ideas for the sequence of Catalan numbers. The audience was comprised primarily of college and university mathematics professors, along with a substantial number of graduate students and undergraduate students, as well as some mathematics teachers from high schools in the Baltimore and Washington, D.C. areas. Since its first presentation, the coverage in this minicourse has expanded over the past 19 years, as I delivered the material nine additional times at later joint national mathematics meetings—the latest being the meetings held in January of 2010 in San Francisco. In addition, the topics have also been presented completely, or in part, at more than a dozen state sectional meetings of the Mathematical Association of America and at several workshops, where, on occasion, some high school students were in attendance. Evaluations provided by those who attended the lectures directed me to further relevant material and also helped to improve the presentations. At all times, the presentations were developed so that everyone in the audience would be able to understand at least some, if not a substantial amount, of the mate- rial. Consequently, this resulting book, which has grown out of these experiences, should be looked upon as an introduction to the many interesting properties, exam- ples, and applications that arise in the study of two of the most fascinating sequences of numbers. As we progress through the various chapters, we should soon come to under- stand why these sequences are often referred to as ubiquitous, especially in courses in discrete mathematics and combinatorics, where they appear so very often. For the Fibonacci numbers, we shall find applications in such diverse areas as set theory, the compositions of integers, graph theory, matrix theory, trigonometry, botany, chem- istry, physics, probability, and computational complexity. We shall find the Catalan numbers arise in situations dealing with lattice paths, graph theory, geometry, partial orders, sequences, pattern avoidance, partitions, computer science, and even sporting events. xi xii PREFACE FEATURES Following are brief descriptions of four of the major features of this book. 1. Useful Resources The book can be used in a variety of ways: (i) As a textbook for an introductory course on the Fibonacci numbers and/or the Catalan numbers. (ii) As a supplement for a course in discrete mathematics or combinatorics. (iii) As a source for students seeking a topic for a research paper or some other type of project in a mathematical area they have not covered, or only briefly covered, in a formal mathematics course. (iv) As a source for independent study. 2. Organization The book is divided into 36 chapters. The first 17 chapters constitute Part One of the book and deal with the Fibonacci numbers. Chapters 18 through 36 comprise Part Two, which covers the material on the Catalan numbers. The two parts can be covered in either order. In Part Two, some references are made to material in Part One. These are usually only comparisons. Should the need arise, one can readily find the material from Part One that is mentioned in conjunction with something covered in Part Two. Furthermore, each of Parts One and Two ends with a bibliography. These references should prove useful for the reader interested in learning even more about either of these two rather amazing number sequences. 3. Detailed Explanations Since this book is to be regarded as an introduction, examples and, especially, proofs are presented with detailed explanations. Such examples and proofs are designed to be careful and thorough. Throughout the book, the presentation is focused primarily on improving understanding for the reader who is seeing most, if not all, of this material for the first time. In addition, every attempt has been made to provide any necessary back- ground material, whenever needed. 4. Exercises There are over 300 exercises throughout the book. These exercises are pri- marily designed to review the basic ideas provided in a given chapter and to introduce additional properties and examples. In some cases, the exercises also extend what is covered in one or more of the chapters. Answers for all the odd-numbered exercises are provided at the back of the book. ANCILLARY There is an Instructor’s Solution Manual that is available for those instructors who adopt this book. The manual can be obtained from the publisher via written request PREFACE xiii on departmental letterhead. It contains the solutions for all the exercises within both parts of the book. ACKNOWLEDGMENTS If space permitted, I should like to thank each of the many participants at the mini- courses, sectional meetings, and workshops, who were so very encouraging over the years.