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How to Count Robert A. Beeler How to Count An Introduction to Combinatorics and Its Applications 2123 Robert A. Beeler Department of Mathematics and Statistics East Tennessee State University Johnson City Tennessee USA ISBN 978-3-319-13843-5 ISBN 978-3-319-13844-2 (eBook) DOI 10.1007/978-3-319-13844-2 Library of Congress Control Number: 2015932250 Springer Cham Heidelberg New York Dordrecht London © Springer International Publishing Switzerland 2015 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) Preface The goal of this book is to provide a reasonably self-contained introduction to com- binatorics. For this reason, this book assumes no knowledge of combinatorics. It does however assume that the reader has been introduced to elementary proof techniques and mathematical reasoning. These modest prerequisites are typically de- veloped at the late sophomore or early junior level. Students wishing to improve their skills in such areas are referred to Mathematical Proofs: A Transition to Advanced Mathematics by Chartrand et al. [14]. This text is aimed at the junior or senior undergraduate level. There is a strong emphasis on computation, problem solving, and proof technique. In particular, there is a particular emphasis on combinatorial proofs for reasons discussed in Sect. 1.6. In addition, this book is written as a “problem based” approach to combinatorics. In each section, specific problems are introduced. Students are then guided in finding the solution to not only the original problem, but a number of variations. Hence, there are a number of examples throughout each section. Often these examples require the student to not only apply the new material, but to implement information developed in previous sections. For this reason, students are generally expected to have a working mastery of the key concepts developed in previous sections before proceeding. In particular, the basic Principle of Inclusion and Exclusion and the Multiplication Principle are used repeatedly. Intuitive descriptions of abstract concepts (such as generating functions) are pro- vided. In addition, supplementary reading on several topics are suggested throughout the text. Hence, this text lends itself not only to a traditional combinatorics course, but also to honors classes or undergraduate research. There are a number of exercises provided at the end of each section. These ex- ercises range from simple computations (in other words, evaluate a formula for a given set of values) to more advanced proofs. Most of the exercises are modeled after examples in the book allowing the student to refer through the text for insight. However, other exercises require deeper problem solving skills. In particular, many of the exercises make use of the key ideas of the Principle of Inclusion and Exclusion and the Multiplication Principle. This helps to reinforce these skills. The first seven chapters form the core of a typical one semester course in com- binatorics. Of these chapters, Sects. 2.6, 2.7, 3.2, and 3.7 are not required for the v vi Preface remainder of the first seven chapters. Instructors wishing to provide a more theo- retical introduction may wish to include Chap. 8 on Pólya theory. In which case, Sect. 2.7 should be covered before introducing this material. Instructors wishing to provide a more applied introduction may wish to sprinkle material on probability from Chap. 9 throughout their course. Instructors may also wish to use the material on combinatorial designs (Chap. 10) to provide more applications. Instructors wish- ing to provide an introduction to graph theory (for instance, in a course on discrete mathematics) may wish to incorporate material from Chap. 11 as well. The author welcomes any constructive suggestions on the improvement of future versions of this text. East Tennessee State University, 2015. Robert A. Beeler, Ph.D., Acknowledgments I would first like to thank my family, D. Beeler, L. Beeler, J. Beeler, and P. Keck for their love and support throughout my life. I would like to thank my colleagues R. Gardner, A. Godbole, T. Haynes, M. Helfgott, D. Knisley, R. Price, and E. Seier for encouraging me to finish this manuscript. Finally, I wish to acknowledge some of the excellent math teachers in my career. In particular, I would like to thank N. Calkin, J. Dydak, R. Jamison, G. Matthews, R. Sharp, C. Wagner, D. Vinson, and J. Xiong. vii Contents 1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 What is Combinatorics? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Induction and Contradiction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.5 The Pigeonhole Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.6 The Method of Combinatorial Proof . . . . . . . . . . . . . . . . . . . . . . . . . 18 2 Basic Counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.1 The Multiplication Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.2 The Addition Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.3 Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.4 Application: Legendre’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.5 Ordered Subsets of [n] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.6 Application: Possible Games of Tic-tac-toe . . . . . . . . . . . . . . . . . . . 44 2.7 Stirling Numbers of the First Kind . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3 The Binomial Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.1 Unordered Subsets of [n] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.2 Application: Hands in Poker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.3 The Binomial Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.4 Identities Involving the Binomial Coefficient . . . . . . . . . . . . . . . . . . 71 3.5 Stars and Bars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.6 The Multinomial Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 3.7 Application: Cryptosystems and the Enigma . . . . . . . . . . . . . . . . . . 89 4 Distribution Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.2 The Solution of Certain Distribution Problems . . . . . . . . . . . . . . . . 97 4.3 Partition Numbers and Stirling Numbers of the Second Kind . . . . 104 4.4 The Twelvefold Way . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 ix x Contents 5 Generating Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 5.1 Review of Factoring and Partial Fractions . . . . . . . . . . . . . . . . . . . . 115 5.2 Review of Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 5.3 Single Variable Generating Functions . . . . . . . . . . . . . . . . . . . . . . . . 127 5.4 Generating Functions with Two or More Variables . . . . . . . . . . . . . 137 5.5 Ordered Words with a Given Set of Restrictions . . . . . . . . . . . . . . . 143 6 Recurrence Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 6.1 Finding Recurrence Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 6.2 The Method of Generating Functions . . . . . . . . . . . . . . . . . . . . . . . . 155 6.3 The Method of Characteristic Polynomials . . . . . . . . . . . . . . . . . . . 163 6.4 The Method of Symbolic Differentiation . . . . . . . . . . . . . . . . . . . . . 174 6.5 The Method of Undetermined Coefficients . . . . . . . . . . . . . . . . . . . 183 7 Advanced Counting—Inclusion and Exclusion . . . . . . . . . . . . . . . . . . . . 195 7.1 The Principle of Inclusion and Exclusion . . . . . . . . . . . . . . . . . . . . . 195 7.2 Items That Satisfy a Prescribed Number of Conditions . . . . . . . . . 205 7.3 Stirling Numbers of the Second Kind and Derangements Revisited 209 7.4 Problème des Ménage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 8 Advanced Counting—Pólya Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 8.1 Equivalence Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 8.2 Group Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 8.3 Burnside’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 8.4 Equivalent Colorings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 8.5 Pólya Enumeration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 9 Application: Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 9.1 Basic Discrete Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 9.2 The Expected Value and Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 9.3 The Binomial Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 9.4 The Geometric Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 9.5 The Poisson Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 9.6 The Hypergeometric Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 10 Application: Combinatorial Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 10.2 Block Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 10.3 Steiner Triple Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 10.4 Finite Projective Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302 11 Application: Graph Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 11.1 What is a Graph? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 11.2 Cycles Within Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 11.3 Planar Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322 11.4 Counting Labeled Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 Contents xi 11.5 Pólya Theory Revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359

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