generatingfunctionology: Third Edition

254 Pages·2005·1.354 MB·English

by  Herbert S. Wilf

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generatingfunctionology
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Third EdiTion

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herbert S. Wilf
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This is the third edition of the highly successful e generatingfunctionology
introduction to the use of generating functions and r
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series in combinatorial mathematics. This new edi- T h i r d E d i T i o n
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tion provides a clear, unified introduction to the basic
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enumerative applications of generating functions. n
g herbert S. Wilf
Special features:
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• Provides applications to the cycle index of the u
symmetric group, permutations and square
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roots, counting polyominoes, and exact covering
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sequences
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• Features an appendix on using Maple® and
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Mathematica® to work with generating functions
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• includes new exercises with complete solutions
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at the end of each chapter
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generatingfunctionology
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generatingfunctionology
Third Edition
Herbert S. Wilf
AKPeters,Ltd.
Wellesley,Massachusetts
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CRC Press
Taylor & Francis Group
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Boca Raton, FL 33487-2742
© 2005 by Taylor & Francis Group, LLC
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No claim to original U.S. Government works
Version Date: 20150311
International Standard Book Number-13: 978-1-4398-6439-5 (eBook - PDF)
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Contents
Preface ix
1 IntroductoryIdeasandExamples 1
1.1 An Easy Two Term Recurrence . . . . . . . . . . . . . . . . 4
1.2 A Slightly Harder Two Term Recurrence . . . . . . . . . . . 5
1.3 A Three Term Recurrence . . . . . . . . . . . . . . . . . . . 9
1.4 A Three Term Boundary Value Problem . . . . . . . . . . . 11
1.5 Two Independent Variables . . . . . . . . . . . . . . . . . . 15
1.6 Another 2-Variable Case . . . . . . . . . . . . . . . . . . . . 18
1.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2 Series 31
2.1 Formal Power Series . . . . . . . . . . . . . . . . . . . . . . 31
2.2 The Calculus of Formal Ordinary Power Series Generating
Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.3 The Calculus of Formal Exponential Generating Functions . 41
2.4 Power Series, Analytic Theory. . . . . . . . . . . . . . . . . 48
2.5 Some Useful Power Series . . . . . . . . . . . . . . . . . . . 55
2.6 Dirichlet Series, Formal theory . . . . . . . . . . . . . . . . 59
2.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3 Cards,Decks,andHands: TheExponentialFormula 77
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.2 Definitions and a Question. . . . . . . . . . . . . . . . . . . 79
3.3 Examples of Exponential Families. . . . . . . . . . . . . . . 80
3.4 The Main Counting Theorems. . . . . . . . . . . . . . . . . 83
3.5 Permutations and Their Cycles . . . . . . . . . . . . . . . . 87
3.6 Set Partitions . . . . . . . . . . . . . . . . . . . . . . . . . . 88
3.7 A Subclass of Permutations . . . . . . . . . . . . . . . . . . 89
3.8 Involutions, etc. . . . . . . . . . . . . . . . . . . . . . . . . . 90
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vi Contents
3.9 2-Regular Graphs . . . . . . . . . . . . . . . . . . . . . . . . 91
3.10 Counting Connected Graphs. . . . . . . . . . . . . . . . . . 92
3.11 Counting Labeled Bipartite Graphs . . . . . . . . . . . . . . 93
3.12 Counting Labeled Trees . . . . . . . . . . . . . . . . . . . . 95
3.13 Exponential Families and Polynomials of ‘Binomial Type.’ . 97
3.14 Unlabeled Cards and Hands . . . . . . . . . . . . . . . . . . 98
3.15 The Money Changing Problem . . . . . . . . . . . . . . . . 102
3.16 Partitions of Integers . . . . . . . . . . . . . . . . . . . . . . 107
3.17 Rooted Trees and Forests . . . . . . . . . . . . . . . . . . . 109
3.18 Historical Notes . . . . . . . . . . . . . . . . . . . . . . . . . 110
3.19 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4 ApplicationsofGeneratingFunctions 115
4.1 Generating Functions Find Averages, etc. . . . . . . . . . . 115
4.2 A Generatingfunctionological View of the Sieve Method . . 117
4.3 The ‘Snake Oil’ Method for Easier Combinatorial Identities 126
4.4 WZ Pairs Prove Harder Identities . . . . . . . . . . . . . . . 138
4.5 Generating Functions and Unimodality, Convexity, etc. . . . 145
4.6 Generating Functions Prove Congruences . . . . . . . . . . 148
4.7 The Cycle Index of the Symmetric Group . . . . . . . . . . 150
4.8 How Many Permutations Have Square Roots? . . . . . . . . 155
4.9 Counting Polyominoes . . . . . . . . . . . . . . . . . . . . . 159
4.10 Exact Covering Sequences . . . . . . . . . . . . . . . . . . . 163
4.11 Waiting for a String . . . . . . . . . . . . . . . . . . . . . . 166
4.12 Blocks of 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
4.13 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
5 AnalyticandAsymptoticMethods 181
5.1 The Lagrange Inversion Formula . . . . . . . . . . . . . . . 181
5.2 Analyticity and Asymptotics (I): Poles . . . . . . . . . . . . 185
5.3 Analyticity and Asymptotics (II): Algebraic Singularities. . 192
5.4 Analyticity and Asymptotics (III): Hayman’s Method . . . 196
5.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
A UsingMapleandMathematica 207
A.1 Series Manipulation . . . . . . . . . . . . . . . . . . . . . . 208
A.2 The RSolve.m Routine . . . . . . . . . . . . . . . . . . . . . 209
A.3 Asymptotics in Maple . . . . . . . . . . . . . . . . . . . . . 211
Solutions 213
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Contents vii
References 239
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Preface
This book is about generating functions and some of their uses in discrete
mathematics. The subject is so vast that I have not attempted to give a
comprehensive discussion. Instead I have tried only to communicate some
of the main ideas.
Generatingfunctionsareabridgebetweendiscretemathematics,onthe
one hand, and continuous analysis (particularly complex variable theory)
ontheother. Itispossibletostudythemsolelyastoolsforsolvingdiscrete
problems. As such there is much that is powerful and magical in the way
generatingfunctionsgiveunifiedmethodsforhandlingsuchproblems. The
reader who wished to omit the analytical parts of the subject would skip
chapter 5 and portions of the earlier material.
To omit those parts of the subject, however, is like listening to a stereo
broadcast of, say, Beethoven’s Ninth Symphony, using only the left audio
channel.
Thefullbeautyofthesubjectofgeneratingfunctionsemergesonlyfrom
tuning in on both channels: the discrete and the continuous. See how they
makethesolutionofdifferenceequationsintochild’splay. Thenseehowthe
theoryoffunctionsofacomplexvariablegives,virtuallybyinspection, the
approximate size of the solution. The interplay between the two channels
is vitally important for the appreciation of the music.
Inrecentyearstherehasbeenavigoroustrendinthedirectionoffinding
bijective proofs of combinatorial theorems. That is, if we want to prove
that two sets have the same cardinality then we should be able to do it by
exhibiting an explicit bijection between the sets. In many cases the fact
thatthetwosetshavethesamecardinalitywasdiscoveredinthefirstplace
by generating function arguments. Also, even though bijective arguments
may be known, the generating function proofs may be shorter or more
elegant.
Thebijectiveproofsgiveoneacertainsatisfyingfeelingthatone‘really’
understands why the theorem is true. The generating function arguments
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