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i i “master” — 2010/9/20 — 12:30 — page i — #1 i i Combinatorics A Guided Tour i i i i i i “master” — 2012/7/26 — 15:22 — page ii — #2 i i (cid:13)c 2010by TheMathematicalAssociationofAmerica(Incorporated) LibraryofCongressCatalogCardNumber2009937059 ElectronicISBN:978-1-61444-607-1 PrintISBN:978-0-88385-762-5 PrintedintheUnitedStatesofAmerica CurrentPrinting(lastdigit): 10987654321 i i i i i i “master” — 2010/9/20 — 12:30 — page iii — #3 i i Combinatorics A Guided Tour David R. Mazur Western New England College ® Publishedanddistributedby TheMathematicalAssociationofAmerica i i i i i i “master” — 2010/9/20 — 12:30 — page iv — #4 i i CommitteeonBooks PaulM.Zorn,Chair MAATextbooksEditorialBoard ZavenA.Karian,Editor GeorgeExner ThomasGarrity CharlesR.Hadlock WilliamHiggins DouglasB.Meade StanleyE.Seltzer ShahriarShahriari KayB.Somers i i i i i i “master” — 2010/9/20 — 12:30 — page v — #5 i i MAATEXTBOOKS CalculusDeconstructed:ASecondCourseinFirst-YearCalculus,ZbigniewH.Nitecki Combinatorics:AGuidedTour,DavidR.Mazur Combinatorics:AProblemOrientedApproach,DanielA.Marcus ComplexNumbersandGeometry,Liang-shinHahn ACourseinMathematicalModeling,DouglasMooneyandRandallSwift CryptologicalMathematics,RobertEdwardLewand DifferentialGeometryanditsApplications,JohnOprea ElementaryCryptanalysis,AbrahamSinkov ElementaryMathematicalModels,DanKalman EssentialsofMathematics,MargieHale FieldTheoryanditsClassicalProblems,CharlesHadlock FourierSeries,RajendraBhatia GameTheoryandStrategy,PhilipD.Straffin GeometryRevisited,H.S.M.CoxeterandS.L.Greitzer GraphTheory:AProblemOrientedApproach,DanielMarcus KnotTheory,CharlesLivingston LieGroups:AProblem-OrientedIntroductionviaMatrixGroups,HarrietPollatsek MathematicalConnections:ACompanionforTeachersandOthers,AlCuoco MathematicalInterestTheory,SecondEdition,LeslieJaneFedererVaalerandJamesW.Daniel MathematicalModelingintheEnvironment,CharlesHadlock MathematicsforBusinessDecisionsPart1:ProbabilityandSimulation(electronictextbook),Richard B.ThompsonandChristopherG.Lamoureux MathematicsforBusinessDecisionsPart2:CalculusandOptimization(electronictextbook),Richard B.ThompsonandChristopherG.Lamoureux TheMathematicsofGamesandGambling,EdwardPackel MathThroughtheAges,WilliamBerlinghoffandFernandoGouvea NoncommutativeRings,I.N.Herstein Non-EuclideanGeometry,H.S.M.Coxeter NumberTheoryThroughInquiry,DavidC.Marshall,EdwardOdell,andMichaelStarbird APrimerofRealFunctions,RalphP.Boas ARadicalApproachtoRealAnalysis,2ndedition,DavidM.Bressoud RealInfiniteSeries,DanielD.BonarandMichaelKhoury,Jr. TopologyNow!,RobertMesserandPhilipStraffin UnderstandingourQuantitativeWorld,JanetAndersenandToddSwanson MAAServiceCenter P.O.Box91112 Washington,DC20090-1112 1-800-331-1MAA FAX:1-301-206-9789 i i i i i i “master” — 2010/9/20 — 12:30 — page vi — #6 i i i i i i i i “master” — 2010/9/20 — 12:30 — page vii — #7 i i Preface Thisbookprovidesa journeythroughintroductorycombinatoricsthatthe reader can un- dertakeduringonesemester,twoquarters,orinaindependentstudyorself-studysetting.It isnotintendedtobeencyclopedic.Rather,itsurveysagoodcross-sectionofcombinatorics as ithas developedwithinthelastcenturywithan eye towardsitscharacteristic brandof thinking,itsinterconnectionswithothermathematicalfields,andsomeofitsapplications. Combinatoricscanrightlybecalledthemathematicsofcounting.Morespecifically,it isthemathematicsoftheenumeration,existence,construction,andoptimizationquestions concerningfinitesets.Herearesomebriefillustrations. Enumeration: How many? How many different 9 9 Sudoku boards are there? (cid:15) (cid:2) Thisnumberhasbeencomputedexactlyanditisastronomical—about6.6sextillion. Determining this number by simply listingevery possible board is not a viable ap- proach.Combinatoricsinvolvesmathematicaltechniquesfordeterminingtheanswer toacountingquestionwithoutlistingtheobjectsbeingcounted. Existence:Isitpossible?Takeany25peoplelivingontheearth.Amongthemembers (cid:15) ofthisgroupwillyoualwaysbeabletofindfourpeoplewhoallknoweachotheror elsefivepeoplewhoalldon’tknoweachother?Yes:thisisguaranteednomatterwhat group of 25 you choose. Despite its innocent-soundingnature, thisquestion wasn’t answereduntil1993andrequiredcarefulcombinatorialanalysisaswellasthousands ofhoursofcomputertime. Construction: Can it be built?The Mariner 9 spacecraft orbitedMars in 1971-72 (cid:15) andsentbackphotographsthatgaveacompletepictureoftheplanet’ssurface. Your CDplayercanplayadiscflawlesslydespiteoccasionalscratchesonthedisc’ssurface. Both of these applications involve error-correcting codes that transmit information with100%accuracy despiteoccasionalerrorsintransmission.Constructionmethods formanyerror-correctingcodesusecombinatorics. Optimization: What is the best way? Your car’s GPS navigation system quickly (cid:15) finds the fastest route from point A to point B. It essentially solves instances of a combinatorial optimization problem called the shortest path problem, which is but oneofabroadclassofnetworkoptimizationproblemsthathavewidespreadmodern application. Inthisbookweconsiderenumeration,existence,andconstructionquestions. Theexamplesaboverightlysuggestthatcombinatoricshasmanymodernapplications. Counting techniques are indispensable in applied probability when the sample space is finite and outcomes are equally likely. Combinatorial design theory grew out of a need vii i i i i i i “master” — 2010/9/20 — 12:30 — page viii — #8 i i viii Preface that statisticianshad inconstructingvalidexperimental designs. Computerscience is re- pletewithapplicationsascombinatorialthinkinginformstheefficiencyofalgorithmsand datastructuresaswellasthecorrectnessofrecursiveprocedures.Linearprogrammingand combinatorial optimizationare fields born from the large-scale logisticalplanningprob- lemsofWorldWarIIandnowinclude,amongmanyothers,applicationstothedesignof transportation and telecommunications networks. Operations research, management science,andindustrialengineeringareotherfieldsinwhichcombinatorialanalysisisusedto solveimportantandpracticalproblems. Beyondspecific examples andproblems,though,thebroaderviewisthatcombinato- rialthinkingisbeneficialandapplicabletomanyareasofmathematics,statistics,computer science, andengineering.Twoofthelargestprofessionalsocieties inthefieldsofmathe- maticsandcomputerscience—the MathematicalAssociationofAmerica (MAA)andthe Association for Computing Machinery (ACM)—recommend that majors and minors in mathematicsandcomputersciencetakecoursesinvolvingagoodamountofdiscretemath- ematicsandcombinatorics. Assuch, combinatoricsisnowproperlyintertwinedwithmodernmathematics. Inthe recent past, combinatoricswas viewedas a usefulsetoftoolsand,at best, asurrogateto otherfields.Nowthatcombinatoricshasgelledintoamorecoherentwhole,itisinteresting toseehowfieldssuchascalculus,analysis,numbertheory,abstractandlinearalgebra,and differentialequationscan be usedas toolstosolvepurelycombinatorialproblems.Some ofthoseresultsaretruemathematicalhighlights. What’s on the tour and what’s not Asmentionedearlier,thisbookprovidesanintroductorysurveyofenumeration,existence, andconstructionquestions.Theemphasisisonenumerationandthefirstfivechapterspro- videthecorematerialoncountingtechniquesandnumberfamilies.Theremainingchapters takeupgraphs,combinatorialdesigns,error-correctingcodes,andpartiallyorderedsets. InChapter1webeginwiththeclassificationandanalysisofbasiccountingquestions. Wealsolaythegroundworkfortherestofourjourneybyintroducingfiveessentialcombi- natorialprinciples:theproductandsumprinciples,thebijectionprinciple,theequivalence principle,andthepigeonholeprinciple.Thelatterisexistential,notenumerative,innature. InChapter2weundertakethestudyofdistributionproblems.Mostcountingquestions are equivalent to questions of counting the ways to distribute “objects” to “recipients.” Throughthesedistributionproblemswemeetseveralmajorplayers:binomialcoefficients, Stirlingnumbers, and integer partitionnumbers. We also introduce and emphasize combinatorialproofs as wellas the technique of recursion:breakingup a large problem into smallersubproblemsofthesametype. In Chapter 3 we introduce inclusion-exclusion, mathematical induction, generating functions,andrecurrencerelations.Thesearealgebraictechniquesincontrasttothecombi- natorialtechniquesofthepreviouschapters.Thecoverageofgeneratingfunctionsincludes techniquesforsolvingrecurrencerelations. In Chapter 4 we use the techniques of the previouschapters to give a more in-depth studyofthebinomialandmultinomialcoefficients, Fibonaccinumbers, Stirlingnumbers ofthefirstandsecondkinds,andintegerpartitionnumbers.Amongotherlinesofinvesti- gation,wederivegeneratingfunctionsforthesefamiliesofnumbers,counttriangulations i i i i i i “master” — 2010/9/20 — 12:30 — page ix — #9 i i Preface ix of the regular n-gon, give combinatorialproofs ofFibonacci number identitiesusingthe ideaoftiling,deriveabeautifulformulafortheBellnumbers,andexploreformulasandan asymptoticestimatefortheintegerpartitionnumbers. InChapter5wecovercountingproblemsinvolvingequivalenceandsymmetryconsid- erations. The main results are the Cauchy-Frobenius-Burnsidetheorem and Po´lya’s enu- merationtheorem. ThoughPo´lya’stheoremarosefromanapplicationtotheenumeration ofchemicalcompounds,ithassinceprovedtobeapowerfulandversatiletoolinallsorts ofotherapplications.We beginthischapter byintroducingthoseaspects ofgrouptheory necessary to understand the theorems, and then give many illustrationsof how to apply them. InChapter6we givea shortsurveyofsome combinatorialproblemsingraphtheory. These include the enumeration oflabeled trees and binary search trees, coloringand the chromatic polynomial, and introductoryRamsey theory. Though Ramsey theory can be introduced without the aid of graphs, the edge-coloring interpretation is convenient and concrete.Thefirstsectionofthischaptercoversbasicgraphtheoryconceptsforthereader whoisunfamiliarwithgraphs. InChapter7wecovertwoofthemostcompellingapplicationsofcombinatorics:com- binatorialdesignsanderror-correctingcodes.Asabonus,themathematicalquestionssur- rounding these applications are just as compelling if not more so. In the three sections on designs we cover existence and construction methods, symmetric designs, and triple systems. In thetwosections onerror-correctingcodes, we constructthe familyofbinary Hamming codes and derive theirerror-correctingproperties, studytheinterplaybetween codes and designs, and discuss the truly astonishing results concerning the existence of perfectcodes. In Chapter 8 we conclude our journey by studyingrelations that are, in some sense, lurkingbehindmuch ofcombinatorics:partiallyorderedsetsor“posets.” We studysome classicalresults(Sperner’stheoremandDilworth’stheorem)andalsotheconceptofposet dimension.InthefinaltwosectionsweintroducethetheoryofMo¨biusinversionanddoso withatwo-foldpurpose:toprovideaunifyingframeworkforseveralcombinatorialideas andtopreparethereaderforfurtherstudy. There are several important topics not included on the tour. The coverage of graph theoryinChapter6,thoughitcontainsanintroductorysection,isfocusedfairlynarrowly onthe topicsmentionedearlier. A majorbranch ofcombinatorics, namely combinatorial optimization,isleftoutentirely.Also,thecoverage ofdesignsandcodes isdrivenbythe particularapplications.Assuch,wedonotcoverprojectiveplanes,combinatorialgeome- tries,orLatinsquares. Features of this book Readingquestions.Whatmakesthisbookaguidedtouraretheapproximately350Ques- tionsspreadthroughouttheeightchapters. These allowthereadertobeanactive partici- pantinthediscussionandaremeanttoprovideamorehonestreflectionoftheprocessby whichwealllearnmathematics.Readingamathbookwithoutpencilandpaperinhandis likestayinginyourhotelandviewingtheinterestingsitesfrom yourwindow.You’llget moreoutofthetourifyouleavethehotelandgoexploreonfoot. i i i i