Algebraic Combinatorics: Walks, Trees, Tableaux, and More

224 Pages·2013·3.891 MB·English

by  Richard P. Stanley

Undergraduate Texts in Mathematics
Undergraduate Texts in Mathematics
SeriesEditors:
SheldonAxler
SanFranciscoStateUniversity,SanFrancisco,CA,USA
KennethRibet
UniversityofCalifornia,Berkeley,CA,USA
AdvisoryBoard:
ColinAdams,WilliamsCollege,Williamstown,MA,USA
AlejandroAdem,UniversityofBritishColumbia,Vancouver,BC,Canada
RuthCharney,BrandeisUniversity,Waltham,MA,USA
IreneM.Gamba,TheUniversityofTexasatAustin,Austin,TX,USA
RogerE.Howe,YaleUniversity,NewHaven,CT,USA
DavidJerison,MassachusettsInstituteofTechnology,Cambridge,MA,USA
JeffreyC.Lagarias,UniversityofMichigan,AnnArbor,MI,USA
JillPipher,BrownUniversity,Providence,RI,USA
FadilSantosa,UniversityofMinnesota,Minneapolis,MN,USA
AmieWilkinson,UniversityofChicago,Chicago,IL,USA
Undergraduate Texts in Mathematics are generally aimed at third- and fourth-
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illustratekeyconceptsaswellasexercisesthatstrengthenunderstanding.
Forfurthervolumes:
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Richard P. Stanley
Algebraic Combinatorics
Walks, Trees, Tableaux, and More
123
RichardP.Stanley
DepartmentofMathematics
MassachusettsInstituteofTechnology
Cambridge,MA,USA
ISSN0172-6056
ISBN978-1-4614-6997-1 ISBN978-1-4614-6998-8(eBook)
DOI10.1007/978-1-4614-6998-8
SpringerNewYorkHeidelbergDordrechtLondon
LibraryofCongressControlNumber:2013935529
MathematicsSubjectClassification(2010):05Exx
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Preface
Thisbookis intendedprimarilyas a one-semesterundergraduatetextfora course
inalgebraiccombinatorics.Themainprerequisitesareabasicknowledgeoflinear
algebra(eigenvalues,eigenvectors,etc.) overa field, existenceof finite fields, and
somerudimentaryunderstandingofgrouptheory.Theone exceptionis Sect.12.6,
whichinvolvesfiniteextensionsoftherationalsincludingalittleGaloistheory.Prior
knowledgeofcombinatoricsisnotessentialbutwillbehelpful.
Why do I write an undergraduate textbook on algebraic combinatorics? One
obvious reason is simply to gather some material that I find very interesting and
hope that students will agree. A second reason concernsstudents who have taken
anintroductoryalgebracourseandwanttoknowwhatcanbedonewiththeirnew-
foundknowledge.Undergraduatecoursesthatrequireabasicknowledgeofalgebra
are typically either advanced algebra courses or abstract courses on subjects like
algebraictopologyandalgebraicgeometry.Algebraiccombinatoricsoffersabyway
off the traditional algebraic highway, one that is more intuitive and more easily
accessible.
Algebraic combinatorics is a huge subject, so some selection process was
necessary to obtain the present text. The main results, such as the weak Erdo˝s–
Moser theorem and the enumerationof de Bruijn sequences,have the feature that
theirstatementdoesnotinvolveanyalgebra.Suchresultsaregoodadvertisements
for the unifying power of algebra and for the unity of mathematics as a whole.
All but the last chapter are vaguely connected to walks on graphs and linear
transformationsrelatedtothem.Thefinalchapterisahodgepodgeofsomeunrelated
elegant applications of algebra to combinatorics. The sections of this chapter are
independent of each other and the rest of the text. There are also three chapter
appendiceson purely enumerative aspects of combinatoricsrelated to the chapter
material:theRSKalgorithm,planepartitions,andtheenumerationoflabelledtrees.
Almost all the material covered here can serve as a gateway to much additional
algebraic combinatorics. We hope in fact that this book will serve exactly this
purpose, that is, to inspire its readers to delve more deeply into the fascinating
interplaybetweenalgebraandcombinatorics.
vii
viii Preface
Many persons have contributed to the writing of this book, but special thanks
should go to Christine Bessenrodt and Sergey Fomin for their careful reading of
portionsofearliermanuscripts.
Cambridge,MA RichardP.Stanley
Contents
Preface............................................................................. vii
BasicNotation.................................................................... xi
1 WalksinGraphs............................................................ 1
2 CubesandtheRadonTransform......................................... 11
3 RandomWalks.............................................................. 21
4 TheSpernerProperty...................................................... 31
5 GroupActionsonBooleanAlgebras ..................................... 43
6 YoungDiagramsandq-BinomialCoefficients .......................... 57
7 EnumerationUnderGroupAction....................................... 75
8 AGlimpseofYoungTableaux............................................. 103
9 TheMatrix-TreeTheorem................................................. 135
10 EulerianDigraphsandOrientedTrees................................... 151
11 Cycles,Bonds,andElectricalNetworks.................................. 163
11.1 TheCycleSpaceandBondSpace................................... 163
11.2 BasesfortheCycleSpaceandBondSpace......................... 168
11.3 ElectricalNetworks .................................................. 172
11.4 PlanarGraphs(Sketch)............................................... 178
11.5 SquaringtheSquare.................................................. 180
12 MiscellaneousGemsofAlgebraicCombinatorics ...................... 187
12.1 The100Prisoners.................................................... 187
12.2 Oddtown.............................................................. 189
12.3 CompleteBipartitePartitionsofK ................................. 190
n
12.4 TheNonuniformFisherInequality.................................. 191
12.5 OddNeighborhoodCovers .......................................... 193
ix
x Contents
12.6 CirculantHadamardMatrices ....................................... 194
12.7 P-RecursiveFunctions............................................... 200
HintsforSomeExercises........................................................ 209
Bibliography...................................................................... 213
Index............................................................................... 219