A Course in Enumeration

496 Pages·2007·4.085 MB·English

by  Martin Aigner

238
Graduate Texts in Mathematics

Editorial Board
S. Axler K.A. Ribet
Graduate Texts in Mathematics

1 TAKEUTI/ZARING. Introduction to 34 SPITZER. Principles of Random Walk.
Axiomatic Set Theory. 2nd ed. 2nd ed.
2 OXTOBY. Measure and Category. 2nd ed. 35 ALEXANDER/WERMER. Several Complex
3 SCHAEFER. Topological Vector Spaces. Variables and Banach Algebras. 3rd ed.
2nd ed. 36 KELLEY/NAMIOKA et al. Linear
4 HILTON/STAMMBACH. A Course in Topological Spaces.
Homological Algebra. 2nd ed. 37 MONK. Mathematical Logic.
5 MAC LANE. Categories for the Working 38 GRAUERT/FRITZSCHE. Several Complex
Mathematician. 2nd ed. Variables.
6 HUGHES/PIPER. Projective Planes. 39 ARVESON. An Invitation to C*-Algebras.
7 J.-P. SERRE. A Course in Arithmetic. 40 KEMENY/SNELL/KNAPP. Denumerable
8 TAKEUTI/ZARING. Axiomatic Set Theory. Markov Chains. 2nd ed.
9 HUMPHREYS. Introduction to Lie Algebras 41 APOSTOL. Modular Functions and
and Representation Theory. Dirichlet Series in Number Theory.
10 COHEN. A Course in Simple Homotopy 2nd ed.
Theory. 42 J.-P. SERRE. Linear Representations of
11 CONWAY. Functions of One Complex Finite Groups.
Variable I. 2nd ed. 43 GILLMAN/JERISON. Rings of Continuous
12 BEALS. Advanced Mathematical Analysis. Functions.
13 ANDERSON/FULLER. Rings and Categories of 44 KENDIG. Elementary Algebraic Geometry.
Modules. 2nd ed. 45 LOÈVE. Probability Theory I. 4th ed.
14 GOLUBITSKY/GUILLEMIN. Stable Mappings 46 LOÈVE. Probability Theory II. 4th ed.
and Their Singularities. 47 MOISE. Geometric Topology in
15 BERBERIAN. Lectures in Functional Analysis Dimensions 2 and 3.
and Operator Theory. 48 SACHS/WU. General Relativity for
16 WINTER. The Structure of Fields. Mathematicians.
17 ROSENBLATT. Random Processes. 2nd ed. 49 GRUENBERG/WEIR. Linear Geometry.
18 HALMOS. Measure Theory. 2nd ed.
19 HALMOS. A Hilbert Space Problem Book. 50 EDWARDS. Fermat’s Last Theorem.
2nd ed. 51 KLINGENBERG. A Course in Differential
20 HUSEMOLLER. Fibre Bundles. 3rd ed. Geometry.
21 HUMPHREYS. Linear Algebraic Groups. 52 HARTSHORNE. Algebraic Geometry.
22 BARNES/MACK. An Algebraic Introduction 53 MANIN. A Course in Mathematical Logic.
to Mathematical Logic. 54 GRAVER/WATKINS. Combinatorics with
23 GREUB. Linear Algebra. 4th ed. Emphasis on the Theory of Graphs.
24 HOLMES. Geometric Functional Analysis 55 BROWN/PEARCY. Introduction to Operator
and Its Applications. Theory I: Elements of Functional Analysis.
25 HEWITT/STROMBERG. Real and Abstract 56 MASSEY. Algebraic Topology: An
Analysis. Introduction.
26 MANES. Algebraic Theories. 57 CROWELL/FOX. Introduction to Knot
27 KELLEY. General Topology. Theory.
28 ZARISKI/SAMUEL. Commutative Algebra. 58 KOBLITZ. p-adic Numbers, p-adic Analysis,
Vol.I. and Zeta-Functions. 2nd ed.
29 ZARISKI/SAMUEL. Commutative Algebra. 59 LANG. Cyclotomic Fields.
Vol.II. 60 ARNOLD. Mathematical Methods in
30 JACOBSON. Lectures in Abstract Algebra I. Classical Mechanics. 2nd ed.
Basic Concepts. 61 WHITEHEAD. Elements of Homotopy
31 JACOBSON. Lectures in Abstract Algebra II. Theory.
Linear Algebra. 62 KARGAPOLOV/MERLZJAKOV. Fundamentals
32 JACOBSON. Lectures in Abstract Algebra III. of the Theory of Groups.
Theory of Fields and Galois Theory. 63 BOLLOBAS. Graph Theory.
33 HIRSCH. Differential Topology.
(continued after index)
Martin Aigner

A Course in

Enumeration

With 55 Figures and 11 Tables

123
Martin Aigner
Freie Universität Berlin
Fachbereich Mathematik und Informatik
Institut für Mathematik II
Arnimallee 3
14195 Berlin, Germany
[email protected]
Editorial Board
S. Axler K.A. Ribet
Mathematics Department Mathematics Department
San Francisco State University University of California, Berkeley
San Francisco, CA 94132 Berkeley, CA 94720-3840
USA USA
[email protected] [email protected]
Library of Congress Control Number: 2007928344
Mathematics Subject Classification (2000): 05-01
ISSN 0072-5285
ISBN 978-3-540-39032-4 Springer Berlin Heidelberg New York
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Preface
Counting things is probably mankind’s earliest mathematical ex-
perience, and so, not surprisingly, combinatorial enumeration oc-
cupies an important place in virtually every mathematical field.
Yet apart from such time-honored notions as binomial coefficients,
inclusion-exclusion, and generating functions, combinatorial enu-
meration is a young discipline. Its main principles, methods, and
fields of application evolved into maturity only in the last century,
and there has been an enormous growth in recent years. The aim
of this book is to give a broad introduction to combinatorial enu-
meration at a leisurely pace, covering the most important subjects
andleadingthereaderinsomeinstancestotheforefrontofcurrent
research.
The text is divided into three parts: Basics, Methods, and Topics.
This should enable the reader to understand what combinatorial
enumeration is all about, to apply the basic tools to almost any
problemheorshemayencounter,andtoproceedtomoreadvanced
methods and some attractive and lively fields of research.
As prerequisites, only the usual courses in linear algebra and cal-
culus and the basic notions of algebra and probability theory are
needed.Sincegraphsareoftenusedto illustrate a particularresult,
it may also be a good thing to have a text on graph theory at hand.
For terminology and notation not listed in the index the books by
R. Diestel, Graph Theory, Springer 2006, and D.B. West, Introduc-
tion to Graph Theory, Prentice Hall 1996, are good sources. Given
these prerequisites, the book is best suited for a senior undergrad-
uate or first-year graduate course.
It is commonplace to stress the importance of exercises. To learn
enumerative combinatorics one simply must do as many exercises
aspossible. Exercisesappearthroughout the textto illustrate some
points and entice the reader to complete proofs or find generali-
zations. There are 666 exercises altogether. Many of them contain
hints, and for those marked with (cid:2) you will find a solution in the
appendix. In each section, the exercises appear in two groups, di-
vided by a horizontal line. Those in the first part should be doable
with modest effort, while those in the second half require a little
VI Preface
more work. Each chapter closes with a special highlight, usually a
famous and attractive problem illustrating the foregoing material,
and a short list of references for further reading.
I am grateful to many colleagues, friends, and students for all
kindsofcontributions.MyspecialthanksgotoMarkdeLongueville,
Jürgen Schütz, and Richard Weiss, who read all or part of the book
in its initial stages; to Margrit Barrett and Christoph Eyrich for the
superb technical work and layout; and to David Kramer for his
meticulous copyediting.
It is my hope that by the choice of topics, examples, and exer-
cisesthebookwillconveysomeoftheintrinsicbeautyandintuitive
mathematical pleasure of the subject.
Berlin, Spring 2007 Martin Aigner
Contents
Introduction................................................. 1
Part I: Basics
1 Fundamental Coefficients ............................... 5
1.1 Elementary Counting Principles ..................... 5
Exercises ........................................... 9
1.2 Subsets and Binomial Coefficients................... 10
Exercises ........................................... 18
1.3 Set-partitions and Stirling Numbers S ............ 20
n,k
Exercises ........................................... 23
1.4 Permutations and Stirling Numbers s ............. 24
n,k
Exercises ........................................... 29
1.5 Number-Partitions .................................. 31
Exercises ........................................... 35
1.6 Lattice Paths and Gaussian Coefficients ............. 36
Exercises ........................................... 42
Highlight: Aztec Diamonds............................... 44
Notes and References.................................... 51
2 Formal Series and Infinite Matrices...................... 53
2.1 Algebra of Formal Series............................ 53
Exercises ........................................... 59
2.2 Types of Formal Series.............................. 60
Exercises ........................................... 65
2.3 Infinite Sums and Products ......................... 66
Exercises ........................................... 70
2.4 Infinite Matrices and Inversion of Sequences ........ 71
Exercises ........................................... 76
2.5 Probability Generating Functions.................... 77
Exercises ........................................... 84
Highlight: The Point of (No) Return....................... 85
Notes and References.................................... 90
VIII Contents
Part II: Methods
3 Generating Functions ................................... 93
3.1 Solving Recurrences ................................ 93
Exercises ........................................... 102
3.2 Evaluating Sums .................................... 105
Exercises ........................................... 110
3.3 The Exponential Formula ........................... 112
Exercises ........................................... 122
3.4 Number-Partitions and Infinite Products ............ 124
Exercises ........................................... 132
Highlight: Ramanujan’s Most Beautiful Formula .......... 136
Notes and References.................................... 141
4 Hypergeometric Summation ............................ 143
4.1 Summation by Elimination.......................... 143
Exercises ........................................... 148
4.2 Indefinite Sums and Closed Forms .................. 148
Exercises ........................................... 155
4.3 Recurrences for Hypergeometric Sums .............. 155
Exercises ........................................... 161
4.4 Hypergeometric Series.............................. 162
Exercises ........................................... 168
Highlight: New Identities from Old ....................... 171
Notes and References.................................... 178
5 Sieve Methods........................................... 179
5.1 Inclusion–Exclusion................................. 179
Exercises ........................................... 189
5.2 Möbius Inversion ................................... 191
Exercises ........................................... 200
5.3 The Involution Principle ............................ 202
Exercises ........................................... 215
5.4 The Lemma of Gessel–Viennot ...................... 217
Exercises ........................................... 229
Highlight: Tutte’s Matrix–Tree Theorem .................. 231
Notes and References.................................... 237
Contents IX
6 Enumeration of Patterns ................................ 239
6.1 Symmetries and Patterns ........................... 239
Exercises ........................................... 248
6.2 The Theorem of Pólya–Redfield ..................... 249
Exercises ........................................... 260
6.3 Cycle Index......................................... 262
Exercises ........................................... 269
6.4 Symmetries on N and R ............................ 270
Exercises ........................................... 276
Highlight: Patterns of Polyominoes....................... 278
Notes and References.................................... 285
Part III: Topics
7 The Catalan Connection................................. 289
7.1 Catalan Matrices and Orthogonal Polynomials....... 290
Exercises ........................................... 297
7.2 Catalan Numbers and Lattice Paths ................. 300
Exercises ........................................... 305
7.3 Generating Functions and Operator Calculus ........ 306
Exercises ........................................... 320
7.4 Combinatorial Interpretation of Catalan Numbers ... 323
Exercises ........................................... 333
Highlight: Chord Diagrams............................... 337
Notes and References.................................... 344
8 Symmetric Functions.................................... 345
8.1 Symmetric Polynomials and Functions .............. 345
Exercises ........................................... 349
8.2 HomogeneousSymmetric Functions ................ 350
Exercises ........................................... 355
8.3 Schur Functions .................................... 356
Exercises ........................................... 366
8.4 The RSK Algorithm ................................. 367
Exercises ........................................... 378
8.5 Standard Tableaux.................................. 380
Exercises ........................................... 383
Highlight: Hook-Length Formulas ........................ 385
Notes and References.................................... 391
X Contents
9 Counting Polynomials................................... 393
9.1 The Tutte Polynomial of Graphs .................... 393
Exercises ........................................... 405
9.2 Eulerian Cycles and the Interlace Polynomial ........ 407
Exercises ........................................... 419
9.3 Plane Graphs and Transition Polynomials ........... 420
Exercises ........................................... 432
9.4 Knot Polynomials................................... 434
Exercises ........................................... 443
Highlight: The BEST Theorem ............................ 445
Notes and References.................................... 449
10 Models from Statistical Physics ......................... 451
10.1 The Dimer Problem and Perfect Matchings .......... 451
Exercises ........................................... 465
10.2 The Ising Problem and Eulerian Subgraphs .......... 467
Exercises ........................................... 480
10.3 Hard Models........................................ 481
Exercises ........................................... 489
10.4 Square Ice .......................................... 490
Exercises ........................................... 504
Highlight: The Rogers–Ramanujan Identities ............. 506
Notes and References.................................... 517
Solutions to Selected Exercises.............................. 519
Chapter 1................................................ 519
Chapter 2................................................ 521
Chapter 3................................................ 524
Chapter 4................................................ 528
Chapter 5................................................ 529
Chapter 6................................................ 533
Chapter 7................................................ 536
Chapter 8................................................ 540
Chapter 9................................................ 544
Chapter 10 .............................................. 547
Notation..................................................... 553
Index........................................................ 557