A Primer in Combinatorics

344 Pages·2021·2.68 MB·English

by  Alexander Kheyfits

AlexanderKheyfits
APrimerinCombinatorics
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Alexander Kheyfits
A Primer in
Combinatorics
|
2nd edition
MathematicsSubjectClassification2020
Primary:05-01;Secondary:97K,91C
Author
AlexanderKheyfits
USA
[email protected]
ISBN978-3-11-075117-8
e-ISBN(PDF)978-3-11-075118-5
e-ISBN(EPUB)978-3-11-075124-6
LibraryofCongressControlNumber:2021940401
BibliographicinformationpublishedbytheDeutscheNationalbibliothek
TheDeutscheNationalbibliothekliststhispublicationintheDeutscheNationalbibliografie;
detailedbibliographicdataareavailableontheInternetathttp://dnb.dnb.de.
©2021WalterdeGruyterGmbH,Berlin/Boston
Typesetting:VTeXUAB,Lithuania
Printingandbinding:CPIbooksGmbH,Leck
www.degruyter.com
Preface to the second edition
Twonewsectionswereaddedtothisedition.Section2.6,“Graphcoloring”,coversthat
subjectinsomedetailand,inparticular,expoundsthefamousfourcolorproblem.Its
current solution requires an essential computer time. We avoid any use of comput-
ers by limiting to the five colors variant, which is done in detail. Another new sec-
tion,Chapter6,“SecondarystructuresoftheRNA”,showshowmodernapplications
ofgraphtheoryleadtonewclassesofgraphs.Wehavealsorefreshedproblemsand
exercisestomanysections.
Acknowledgment
ThiseditionwasmadepossibleduetomanyprofessionaleffortsofStevenElliot,Ute
Skambraks,VilmaVaičeliūnienėandtheentirestaffofDeGruyter,towhomtheauthor
isverythankful.
https://doi.org/10.1515/9783110751185-201

Preface to the first edition
Combinatorialanalysisorcombinatorics,forshort,dealswithenumerativeproblems
whereonemustanswerthequestion“Howmany?”or“Inhowmanyways?”Other
problemsareconcernedwiththeexistenceofcertaincombinatorialobjectssubjectto
variousconstraints.Thesekindsofproblemsareconsideredinthisbook.
Combinatorialproblems,methodsandgraphicalmodelsareabundantinmany
areasrangingfromengineeringandfinancialsciencetohumanitariandisciplineslike
sociology,psychology,medicineandsocialsciences,nottomentionmathematicsand
computerscience.Aspartsofdiscretemathematics,combinatoricsandgraphtheory
havebecomeindispensablepartsofintroductoryandadvancedmathematicaltrain-
ingforeveryonedealingnotonlywithquantitativebutalsowithqualitativedata.
Moreover, combinatorics and graph theory have a remarkable and uncommon
feature—tobeginitsstudy,oneneedsnobackgroundbutelementaryalgebraandcom-
monsense.Evensimplecombinatorialproblemsoftenleadtointeresting,sometimes
difficultquestionsandallowaninstructortointroducevariousimportantmathemat-
icalideasandconceptsandtoshowthenatureofmathematicalreasoningandproof.
Thesequalitiesmakecombinatoricsandgraphtheoryanexcellentchoiceforanin-
troductorymathematicalclassforstudentsofanyage,levelandmajor.
Thisisatextforaone-semestercourseincombinatoricswithelementsofgraph
theory.Itcanbeusedintwomodes.Thefirstthreechapterscoveranintroductoryma-
terialandcanbe(andhaveactuallybeen)usedforanundergraduateclassincombi-
natoricsand/ordiscretemathematics,aswellasforaproblem-solvingseminaraimed
atundergraduateandevenmotivatedhigh-schoolstudents.
Chapters4and5areofmoreadvancedlevelandthewholebookincludesenough
materialforanentry-levelgraduatecourseincombinatorics.Forthemathematically
inclinedreader,thematerialhasbeendevelopedsystematicallyandincludesallthe
proofs.Afterthisbook,thereadercanstudymoreadvancedcourses,e.g.[1,9,10,22,
51].Atthesametime,thereaderwhoisprimarilyinterestedinapplyingcombinatorial
methodscanskip(mostof)theproofsandconcentrateonproblemsandmethodsof
theirsolution.
In Chapter 1 we introduce basic combinatorial concepts, such as the sum and
productrules,combinations,permutations,andarrangementswithandwithoutrep-
etition.Variousparticularelementarymethodsofsolvingcombinatorialproblemsare
alsoconsideredthroughoutthebook,suchas,forinstance,thetrajectorymethodin
Section1.4orFerrersdiagramsinSection4.4.InSection1.6weapplythemethodsof
Sections1.1–1.5todeveloptheelementaryprobabilitytheoryforrandomexperiments
withfinitesamplespaces.Ourgoalinthissectionisnottogiveasystematicexpo-
sitionofprobabilitytheory,butrathertoshowsomemeaningfulapplicationsofthe
combinatorialmethodsdevelopedearlier.
Chapter2containsanintroductiontographtheory.Aftersettingupthebasicvo-
cabularyinSections2.1–2.2,inthenextthreesectionswestudypropertiesoftrees,
https://doi.org/10.1515/9783110751185-202
VIII | Prefacetothefirstedition
Eulerianandplanargraphs,andsomeproblemsofgraphcoloringandgraphicalenu-
meration.ManyothergraphtheoryproblemsappearinChapters3–5.Asanapplica-
tionofthemethodsdevelopedinChapters1–2,inChapter3wegiveanelementaryin-
troductiontohierarchicalclusteringalgorithms.Thistopichaslikelyneverappeared
intextbooksbefore.
Chapter4isdevotedtomoreadvancedmethodsofenumerativecombinatorics.
Sections 4.1–4.2 cover inversion formulas, including the Möbius inversion, and the
PrincipleofInclusion–Exclusion.Themethodofgeneratingfunctionsisdeveloped
inSection4.3.Generatingfunctionsareintroducedasanalyticalobjects,thesumsof
convergingpowerseries.InSection4.4weconsiderseveralapplicationsofthemethod
ofgeneratingfunctions,inparticularpartitionsandcompositionsofintegernumbers
andlinearrecurrencerelations(differenceequations)withconstantcoefficients.The
Pólya–RedfieldenumerationtheoryisconsideredinSection4.5.
Thelastchapterofthebookisconcernedwithcombinatorialexistenceproblems.
TheRamseytheoremanditsapplicationsareconsideredinSection5.1.TheDirich-
let (pigeonhole) principle follows immediately. Section 5.2 treats Hall’s theorem on
systemsofdistinctrepresentatives(themarriageproblem)andsomeofitsequivalent
statements, namely, König’s theorem on zero-one matrices and Dilworth’s theorem
onchainsinpartiallyorderedsets.Anexampleofanextremalcombinatorialproblem
(theassignmentproblem)isalsoconsideredhere.
Section5.3containsanintroductiontothetheoryofbalancedblockdesigns.We
consideronlyrecursivemethodsofconstructionofblockdesignssincedeepalgebraic
resultsarebeyondthescopeofthisbook.Finally,Section5.4isdevotedtothesystems
oftriplesconcludingwiththeproof,duetoHilton[30]ofthenecessaryandsufficient
conditionsoftheexistenceofSteiner’striplesystems.
Theauthor’scredointeachingmathematicsinvolvesadvancingfromexamples
andmodelproblemstotheoryandthenbacktoproblemsolving.Thisapproachworks
especiallywellincombinatorics.Everysectionofthebookstartswithsimplemodel
problems.Discussingandsolvingtheseproblems,wederivethebasicconceptsand
definitions.Then,westudyessentialpropertiesoftheconceptsdevelopedandagain
solveproblemstoillustratetheideas,methods,andtheirapplications.Inparticular,
somepartsofproofsareleftasproblemstobesolvedbythereader.Studyingtheso-
lutionsoftypicalproblemsinthebook,thereadercanquicklygraspthemethodsof
solvingvariouscombinatorialproblemsandapplythesemethodstoarangeofsimilar
problemsinanysubject.Thusthebookcanbeusedasaself-studyguidebythereader
interestedinsolvingcombinatorialproblems.
Morethan800problemsconstituteanintegralpartofthetext.Manyproblems
aredrawnfromliterature,somearefolklore,andsomeareoriginal.Manyproblems
aresolvedinthetext,scoresofotherproblemsandexercisesareintheendofeach
section.Additionalproblemscanbefoundinthebookscitedinthelistofreferences,
specifically,in[11,13,29,38,39,53].Interestingtopicsforfurtherreadingandindivid-
Prefacetothefirstedition | IX
ualprojectscanbefoundin[4].Solutions,answersorhintstoselectedproblemsand
exercisesaregivenintheendofthebook.
Combinatorial problems often provide natural intuitive motivation and models
forimportantmathematicalideasandconcepts,suchasoperationsonsets,various
classesoffunctions,classesofbinaryrelations,andmanyothers.Primarycombina-
torial concepts, permutations, combinations and alike, can be naturally defined in
termsofsettheoryoperationsandfunctions.Inthetext,wesystematicallyusethis
approachthatcanbetraced(atleast)asfarbackasC.Berge’smonograph[8].Notto
mentionitsconcisenessandtheoreticalmerits,thisset-theorybasedapproachisof-
tenadvantageousinproblemsolving,andwedemonstratethisinthetextusingmany
examples.Thisapproachremovestheambiguitythatisoftenpresentincombinato-
rialproblems,especiallywhendifferentobjectsmustbeidentified,andsignificantly
reducesthenumberofstudenterrors.
Itistheauthor’sexperiencethatfreshmenusuallymasterthisapproachwithease
andsuccessfullyapplyittoproblemsolving.Forthereaderunfamiliarwiththelan-
guageandbasicsofsettheory,Section1.1systematicallydevelopssomestandardter-
minology,whichisusedinthefollowingsections.Thereaderfamiliarwithnaiveset
theorycanskipSection1.1andreferbacktoitasneeded.
Veryfewnon-elementaryconceptsareincludedinthetext.Noconceptbeyond
theprecalculuslevelappearsbeforeSection4.3.Twocalculus-levelconcepts,those
ofderivativesofelementaryfunctionsandofconvergingseries,appearinSection4.3
ongeneratingfunctions.Fromthispointonthebookcanbesubtitled“Combinatorics
throughtheeyesofananalyst”.Eventhenotionofaconvergingseriescanbeelim-
inatedandreplacedbythefinitaryconceptofgeneratingpolynomials,thatis,trun-
cated power series, and we solve a few problems to demonstrate the method. This
approachmakesthemethodofgeneratingfunctionsaccessibletothereaderwithout
anycalculusbackgroundatall,thoughcalculationsbecomemoretedious.
Itshouldbenotedthatthesedaysmanycollegestudentstakeatleastonecalculus
class,butafterwardstheyseenoactualapplicationofcalculus.Therefore,somenon-
trivialexamplesofapplicationsofcalculusideasandmethodsareappropriate.The
samecanbesaidofthefewelementaryalgebraicconcepts(groups,rings)appearing
inChapters4and5.
The book is self-contained; all the concepts and definitions used are defined
and explained by examples. The Index includes references to important groups of
problems and specific methods of their solution, such as “coloring problems” or
“methodofgeneratingfunctions”.Throughoutthetext,weuseseveralabbreviations:
GFstandsforgeneratingfunction(s),EGFforexponentialgeneratingfunction(s),SDR
forsystem(s)ofdistinctrepresentatives,andBIBDforbalancedincompleteblockde-
sign(s).Theorems,lemmas,problems,etc.,havethree-digitnumbering,thus,Prob-
lem1.2.3referstothethirdprobleminthetextofSection1.2ofChapter1,while1.2.3
meansExercise1.2.3intheendofSection1.2.Figureshavetwo-digitnumbering,thus
X | Prefacetothefirstedition
◻
Fig.2.3referstothethirdfigureinChapter2.Thesymbol indicatestheendsofthe
proofsofstatementsorsolutionsofproblems.
Combinatorialproblemsandgraphicalmodelshavebeenstudiedbymanyout-
standing scientists for thousands of years. The web site www.degruyter.com of de
GruyterGmbHcontainsmanyinterestinglinksdescribingthehistoryofthesedevel-
opmentsandlivesofthepeopleinvolved.Thecoffeecupicon indicatesthatthereis
informationavailableatthewebsite.Anyremarks,correctionsandsuggestionsabout
[email protected].
Acknowledgments
Chapter3isarevisedversionofModule03-1intheDIMACSseriesofeducationalmod-
ules, written when the author participated in Reconnect 1998 and Reconnect 1999
conferencesattheDIMACSCenteratRutgersUniversityofNewJersey.Theauthoris
gratefultotheDIMACSCenter,itsDirectorProfessorFredRobertsandProfessorMelvin
JanowitzfortheirhospitalityandthekindpermissiontoincludeModule03-1inthis
text,andtoProfessorCatherineMcGeochforhergeneroushelp.
Itisfinallytheauthor’sgreatpleasuretothankSimonAlbroscheit,RobertPlato,
FriederikeDittbernerandthestaffofdeGruyterGmbHfortheirfriendlyandhighly
professionalhandlingofthewholepublishingprocess.