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Combinatorics Second Edition WILEY-INTERSCIENCE SERIESINDISCRETE MATHEMATICS ANDOPTIMIZATION ADVISORYEDITORS RONALDL.GRAHM UniversityofCaliforniaatSanDiego,U.S.A. JANKARELLENSTRA DepartmentofMathematicsandComputerScience, EindhovenUniversityofTechnology,Eindhoven,TheNetherlands JOELH.SPENCER CourantInstitute,NewYork,NewYork,U.S.A. Acompletelistoftitlesinthisseriesappearsattheendofthisvolume. Combinatorics SECOND EDITION RUSSELL MERRIS California State University, Hayward A JOHN WILEY & SONS, INC., PUBLICATION Thisbookisprintedonacid-freepaper.(cid:2)1 Copyright#2003byJohnWiley&Sons,Inc.,Hoboken,NewJersey.Allrightsreserved. PublishedsimultaneouslyinCanada. Nopartofthispublicationmaybereproduced,storedinaretrievalsystemortransmittedinany formorbyanymeans,electronic,mechanical,photocopying,recording,scanningorotherwise,exceptas permittedunderSections107or108ofthe1976UnitedStatesCopyrightAct,withouteithertheprior writtenpermissionofthePublisher,orauthorizationthroughpaymentoftheappropriateper-copyfee totheCopyrightClearanceCenter,222RosewoodDrive,Danvers,MA01923,(978)750-8400, fax(978)750-4744.RequeststothePublisherforpermissionshouldbeaddressedtothePermissions Department,JohnWiley&Sons,Inc.,605ThirdAvenue,NewYork,NY10158-0012, (212)850-6011,fax:(212)850-6008,E-Mail:[email protected]. Fororderingandcustomerservice,call1-800-CALL-WILEY. LibraryofCongressCataloging-in-PublicationData: Merris,Russell,1943– Combinatorics/RussellMerris.–2nded. p. cm.–(Wileyseriesindiscretemathematicsandoptimization) Includesbibliographicalreferencesandindex. ISBN-0-471-26296-X(acid-freepaper) 1.Combinatorialanalysis I.Title.II.Series. QA164.M472003 5110.6–dc21 2002192250 PrintedintheUnitedStatesofAmerica. 10 9 8 7 6 5 4 3 2 1 This book is dedicated to my wife, Karen Diehl Merris. Contents Preface ix Chapter 1 The Mathematics of Choice 1 1.1. The Fundamental Counting Principle 2 1.2. Pascal’s Triangle 10 *1.3. Elementary Probability 21 *1.4. Error-Correcting Codes 33 1.5. Combinatorial Identities 43 1.6. Four Ways to Choose 56 1.7. The Binomial and Multinomial Theorems 66 1.8. Partitions 76 1.9. Elementary Symmetric Functions 87 *1.10. Combinatorial Algorithms 100 Chapter 2 The Combinatorics of Finite Functions 117 2.1. Stirling Numbers of the Second Kind 117 2.2. Bells, Balls, and Urns 128 2.3. The Principle of Inclusion and Exclusion 140 2.4. Disjoint Cycles 152 2.5. Stirling Numbers of the First Kind 161 Chapter 3 Po´lya’s Theory of Enumeration 175 3.1. Function Composition 175 3.2. Permutation Groups 184 3.3. Burnside’s Lemma 194 3.4. Symmetry Groups 206 3.5. Color Patterns 218 3.6. Po´lya’s Theorem 228 3.7. The Cycle Index Polynomial 241 Note:Asterisksindicateoptionalsectionsthatcanbeomittedwithoutlossofcontinuity. vii viii Contents Chapter 4 Generating Functions 253 4.1. Difference Sequences 253 4.2. Ordinary Generating Functions 268 4.3. Applications of Generating Functions 284 4.4. Exponential Generating Functions 301 4.5. Recursive Techniques 320 Chapter 5 Enumeration in Graphs 337 5.1. The Pigeonhole Principle 338 *5.2. Edge Colorings and Ramsey Theory 347 5.3. Chromatic Polynomials 357 *5.4. Planar Graphs 372 5.5. Matching Polynomials 383 5.6. Oriented Graphs 394 5.7. Graphic Partitions 408 Chapter 6 Codes and Designs 421 6.1. Linear Codes 422 6.2. Decoding Algorithms 432 6.3. Latin Squares 447 6.4. Balanced Incomplete Block Designs 461 Appendix A1 Symmetric Polynomials 477 Appendix A2 Sorting Algorithms 485 Appendix A3 Matrix Theory 495 Bibliography 501 Hints and Answers to Selected Odd-Numbered Exercises 503 Index of Notation 541 Index 547 Preface Thisbookisintendedtobeusedasthetextforacourseincombinatoricsatthelevel of beginning upper division students. It has been shaped by two goals: to make somefairlydeepmathematicsaccessible tostudentswithawiderangeofabilities, interests,andmotivationsandtocreateapedagogicaltoolusefultothebroadspec- trum of instructors who bring a variety of perspectives and expectations to such a course. The author’s approach to the second goal has been to maximize flexibility. Following a basic foundation in Chapters 1 and 2, each instructor is free to pick andchoose themostappropriatetopics fromthe remainingfourchapters.Assum- marizedinthechartbelow,Chapters3–6arecompletelyindependentofeachother. Flexibility is further enhanced by optional sections and appendices, by weaving some topics into the exercise sets of multiple sections, and by identifying various pointsofdeparturefromeachofthefinalfourchapters.(Thepriceofthisflexibility issomeredundancy,e.g.,severaldefinitionscanbefoundinmorethanoneplace.) Chapter 1 Chapter 2 Chapter 5 Chapter 3 Chapter 4 Chapter 6 Turning to the first goal, students using this book are expected to have been exposed to, even if they cannot recall them, such notions as equivalence relations, partialfractions,theMaclaurinseriesexpansionforex,elementaryrowoperations, determinants,andmatrixinverses.Acoursedesignedaroundthisbookshouldhave as specific prerequisites those portions of calculus and linear algebra commonly found among the lower division requirements for majors in the mathematical and computer sciences. Beyond these general prerequisites, the last two sections of Chapter5presumethereadertobefamiliarwiththedefinitionsofclassicaladjoint ix