Table Of ContentCombinatorics: AncientandModern
TitlepageofAthanasiusKircher’sArsMagnaSciendiSiveCombinatoria(TheGreatArtof
Knowledge,ortheCombinatorialArt)of1669.
COMBINATORICS:
ANCIENT AND
MODERN
Editedby
robin wilson
and
john j. watkins
1
3
GreatClarendonStreet,Oxford,OX26DP,
UnitedKingdom
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FOREWORD
O
ne of the most compelling instincts that human beings have is the
irresistibleurgetolookforpatterns:thisisapparentfromtheearli-
estattemptsofourancestorstounderstandtheworldaroundthem.
Mathematicshasoftenbeendescribedasthescienceofpatterns,andperhaps
morethananyothermathematicalfield,thisrepresentstheheartandsoulof
combinatorics.
In this masterful volume, the editors have brought together a wonderful
andfar-rangingcollectionofchaptersbydistinguishedauthors.Theseaccounts
surveythesubjectofcombinatorics,beginningwiththeearliestwrittenresults
in the subject, continuing with its development in a variety of cultures, such
as Indian, Chinese, Islamic, andJewish, andprogressing to theemergence of
whatwenowthinkofasmoderncombinatorics.Fromtheintroductorychapter,
‘Two thousand years of combinatorics’, by Donald Knuth to Peter Cameron’s
‘A personal view of combinatorics’, the book covers a wide range of topics
and offers to both the novice in the subject and to the experts a full plate of
interestingfactsandviewpoints.Thisisthefirsttimethatsuchacompilation
hasbeenattemptedand,intheopinionofthisreader,itsucceedsbrilliantly.
RonaldGraham
FormerPresidentoftheAmericanMathematicalSociety
andtheMathematicalAssociationofAmerica
foreword | v
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PREFACE
WhofirstpresentedPascal’striangle?(ItwasnotPascal.)
WhofirstpresentedHamiltoniangraphs?(ItwasnotHamilton.)
WhofirstpresentedSteinertriplesystems?(ItwasnotSteiner.)
Misattributionsappearthroughoutthehistoryofmathematics,andcombina-
torics has its share. Frequent errors in historical accuracy are perpetrated by
thosewhowouldbeashamedtoallowsucherrorsintheirmathematicalwrit-
ings:notableexamplesarethecommonassertionsthatEulerdrewafour-vertex
graphtosolvetheKönigsbergbridgesproblemandthatDescartesdiscovered
Euler’spolyhedronformula(theydidn’t).
Today the history of mathematics is a well-studied and vibrant area of
research,withbooksandscholarlyarticlespublishedonvariousaspectsofthe
subject.Yet,thehistoryofcombinatoricsseemstohavebeenlargelyoverlooked:
manycombinatorialistsseemuninterestedinthehistoryoftheirsubject,while
historiansofmathematicshavetendedtobypassthefascinationsofcombina-
torics.Itisourhopethatthisbook,writtenbynoteworthyexpertsinthearea,
willgosomewaytoredressthis.
Thisbookservestwopurposes:
• It constitutes what is perhaps the first book-length survey of the history of
combinatorics.
• It assembles,for the first time in a single source,research on the history of
combinatoricsthatwouldotherwisebeinaccessibletothegeneralreader.
Thechaptershavebeencontributedbysixteenexperts,withtopicscorrespond-
ingtotheirparticularareasofresearch.Someofthisresearchreceivesitsfirst
airinghere,whileotherchaptersarebasedonworkthathasappearedelsewhere
butislargelyunavailabletothosewithoutaccesstoresearchjournalsorlarge
preface | vii
universitylibraries.Inordertomakethebookeasiertoread,wehaveendeav-
ouredtostandardizethestylethroughoutthebook,andwearegratefultothe
authorsfortheirforbearancewithusasweproposedchangestotheirdraftsand
imposedourstylisticconventions.
The opening section is an introduction to two thousand years of combi-
natorics, adapted with permission from a section of Volume 4 of Donald E.
Knuth’scelebratedmulti-volumeworkTheArtofComputerProgramming.This
is followed by seven chapters on early combinatorics, leading from Indian
andChinesewritingsonpermutationstolate-Renaissancepublicationsonthe
arithmeticaltriangle.Thenextsevenchapterstracethesubsequentstory,from
Euler’scontributionstosuchwide-rangingtopicsaspartitions,polyhedra,and
latinsquarestothe20th-centuryadvancesincombinatorialsettheory,enumer-
ation,andgraphtheory.Thebookconcludeswithsomecombinatorialreflec-
tionsbythedistinguishedcombinatorialistPeterJ.Cameron.
Naturally,asthefirstbookofthiskind,thisvolumecannothopetobecom-
prehensive,andyouwillnoticetopicsthataremissingoronlyminimallydis-
cussed:theseincludecombinatorialoptimization,combinatorialidentities,and
recreationalcombinatorics.Whilethesesubjects(andothers)wereconsidered
forinclusionwefeltthatthenecessaryconstraintsofavolumeofmanageable
scopeandsizeleft toolittleroom.Suchomissionsnotwithstanding,we hope
thatasanoverviewofrecentandongoinghistoricalworkthisbookprovides
usefulbackgroundinformationandinspirationforfutureresearch.
As with many edited volumes, this book is not intended to be read from
covertocover,althoughitcanbe.Rather,itisintendedtoserveasavaluable
resource to a variety of audiences. We hope that combinatorialists with little
ornoknowledgeaboutthedevelopmentoftheirsubjectwillfindthehistorical
treatmentstimulating,andthatthespecialisthistorianofmathematicswillview
itsassortedsurveysasanencouragementforfurtherresearchincombinatorics.
For the more general reader, we hope that it provides an introduction to a
fascinatingandtoolittleknownsubjectthatcontinuestostimulateandinspire
theworkofscholarstoday.
Finally, we’d like to express our thanks to Keith Mansfield, Viki Mortimer,
and Clare Charles at Oxford University Press, and also to our host institu-
tions, Pembroke College, Oxford University; The Open University, UK; and
TheColoradoCollege,USA.
TheEditors
April2013
viii | preface
CONTENTS
PartI Introduction
Twothousandyearsofcombinatorics 3
Donald E. Knuth
PartII AncientCombinatorics
1. Indiancombinatorics 41
Takanori Kusuba and Kim Plofker
2. China 65
Andrea Bréard
3. Islamiccombinatorics 83
Ahmed Djebbar
4. Jewishcombinatorics 109
Victor J. Katz
5. Renaissancecombinatorics 123
Eberhard Knobloch
6. Theoriginsofmoderncombinatorics 147
Eberhard Knobloch
7. Thearithmeticaltriangle 167
A. W. F. Edwards
PartIII ModernCombinatorics
8. Earlygraphtheory 183
Robin Wilson
contents | ix
Description:Who first presented Pascal's triangle? (It was not Pascal.)Who first presented Hamiltonian graphs? (It was not Hamilton.)Who first presented Steiner triple systems? (It was not Steiner.) The history of mathematics is a well-studied and vibrant area of research, with books and scholarly articles publ
