How to Count: An Introduction to Combinatorics and Its Applications

368 Pages·2015·4.91 MB·English

by  Robert A. Beeler (auth.)

How to Count
Robert A. Beeler
How to Count
An Introduction to Combinatorics and Its
Applications
2123
Robert A. Beeler
Department of Mathematics and Statistics
East Tennessee State University
Johnson City
Tennessee
USA
ISBN 978-3-319-13843-5 ISBN 978-3-319-13844-2 (eBook)
DOI 10.1007/978-3-319-13844-2
Library of Congress Control Number: 2015932250
Springer Cham Heidelberg New York Dordrecht London
© Springer International Publishing Switzerland 2015
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Preface
The goal of this book is to provide a reasonably self-contained introduction to com-
binatorics. For this reason, this book assumes no knowledge of combinatorics.
It does however assume that the reader has been introduced to elementary proof
techniques and mathematical reasoning. These modest prerequisites are typically de-
veloped at the late sophomore or early junior level. Students wishing to improve their
skills in such areas are referred to Mathematical Proofs: A Transition to Advanced
Mathematics by Chartrand et al. [14].
This text is aimed at the junior or senior undergraduate level. There is a strong
emphasis on computation, problem solving, and proof technique. In particular, there
is a particular emphasis on combinatorial proofs for reasons discussed in Sect. 1.6.
In addition, this book is written as a “problem based” approach to combinatorics. In
each section, specific problems are introduced. Students are then guided in finding
the solution to not only the original problem, but a number of variations. Hence, there
are a number of examples throughout each section. Often these examples require the
student to not only apply the new material, but to implement information developed in
previous sections. For this reason, students are generally expected to have a working
mastery of the key concepts developed in previous sections before proceeding. In
particular, the basic Principle of Inclusion and Exclusion and the Multiplication
Principle are used repeatedly.
Intuitive descriptions of abstract concepts (such as generating functions) are pro-
vided. In addition, supplementary reading on several topics are suggested throughout
the text. Hence, this text lends itself not only to a traditional combinatorics course,
but also to honors classes or undergraduate research.
There are a number of exercises provided at the end of each section. These ex-
ercises range from simple computations (in other words, evaluate a formula for a
given set of values) to more advanced proofs. Most of the exercises are modeled
after examples in the book allowing the student to refer through the text for insight.
However, other exercises require deeper problem solving skills. In particular, many
of the exercises make use of the key ideas of the Principle of Inclusion and Exclusion
and the Multiplication Principle. This helps to reinforce these skills.
The first seven chapters form the core of a typical one semester course in com-
binatorics. Of these chapters, Sects. 2.6, 2.7, 3.2, and 3.7 are not required for the
v
vi Preface
remainder of the first seven chapters. Instructors wishing to provide a more theo-
retical introduction may wish to include Chap. 8 on Pólya theory. In which case,
Sect. 2.7 should be covered before introducing this material. Instructors wishing to
provide a more applied introduction may wish to sprinkle material on probability
from Chap. 9 throughout their course. Instructors may also wish to use the material
on combinatorial designs (Chap. 10) to provide more applications. Instructors wish-
ing to provide an introduction to graph theory (for instance, in a course on discrete
mathematics) may wish to incorporate material from Chap. 11 as well.
The author welcomes any constructive suggestions on the improvement of future
versions of this text.
East Tennessee State University, 2015. Robert A. Beeler, Ph.D.,
Acknowledgments
I would first like to thank my family, D. Beeler, L. Beeler, J. Beeler, and P. Keck
for their love and support throughout my life. I would like to thank my colleagues
R. Gardner, A. Godbole, T. Haynes, M. Helfgott, D. Knisley, R. Price, and E. Seier
for encouraging me to finish this manuscript. Finally, I wish to acknowledge some
of the excellent math teachers in my career. In particular, I would like to thank N.
Calkin, J. Dydak, R. Jamison, G. Matthews, R. Sharp, C. Wagner, D. Vinson, and J.
Xiong.
vii
Contents
1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 What is Combinatorics? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Induction and Contradiction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.5 The Pigeonhole Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.6 The Method of Combinatorial Proof . . . . . . . . . . . . . . . . . . . . . . . . . 18
2 Basic Counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.1 The Multiplication Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2 The Addition Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.3 Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.4 Application: Legendre’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.5 Ordered Subsets of [n] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.6 Application: Possible Games of Tic-tac-toe . . . . . . . . . . . . . . . . . . . 44
2.7 Stirling Numbers of the First Kind . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3 The Binomial Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.1 Unordered Subsets of [n] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.2 Application: Hands in Poker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.3 The Binomial Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.4 Identities Involving the Binomial Coefficient . . . . . . . . . . . . . . . . . . 71
3.5 Stars and Bars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.6 The Multinomial Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.7 Application: Cryptosystems and the Enigma . . . . . . . . . . . . . . . . . . 89
4 Distribution Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.2 The Solution of Certain Distribution Problems . . . . . . . . . . . . . . . . 97
4.3 Partition Numbers and Stirling Numbers of the Second Kind . . . . 104
4.4 The Twelvefold Way . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
ix
x Contents
5 Generating Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.1 Review of Factoring and Partial Fractions . . . . . . . . . . . . . . . . . . . . 115
5.2 Review of Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.3 Single Variable Generating Functions . . . . . . . . . . . . . . . . . . . . . . . . 127
5.4 Generating Functions with Two or More Variables . . . . . . . . . . . . . 137
5.5 Ordered Words with a Given Set of Restrictions . . . . . . . . . . . . . . . 143
6 Recurrence Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
6.1 Finding Recurrence Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
6.2 The Method of Generating Functions . . . . . . . . . . . . . . . . . . . . . . . . 155
6.3 The Method of Characteristic Polynomials . . . . . . . . . . . . . . . . . . . 163
6.4 The Method of Symbolic Differentiation . . . . . . . . . . . . . . . . . . . . . 174
6.5 The Method of Undetermined Coefficients . . . . . . . . . . . . . . . . . . . 183
7 Advanced Counting—Inclusion and Exclusion . . . . . . . . . . . . . . . . . . . . 195
7.1 The Principle of Inclusion and Exclusion . . . . . . . . . . . . . . . . . . . . . 195
7.2 Items That Satisfy a Prescribed Number of Conditions . . . . . . . . . 205
7.3 Stirling Numbers of the Second Kind and Derangements Revisited 209
7.4 Problème des Ménage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
8 Advanced Counting—Pólya Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
8.1 Equivalence Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
8.2 Group Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
8.3 Burnside’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
8.4 Equivalent Colorings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
8.5 Pólya Enumeration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
9 Application: Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
9.1 Basic Discrete Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
9.2 The Expected Value and Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
9.3 The Binomial Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
9.4 The Geometric Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276
9.5 The Poisson Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
9.6 The Hypergeometric Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
10 Application: Combinatorial Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
10.2 Block Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294
10.3 Steiner Triple Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
10.4 Finite Projective Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
11 Application: Graph Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
11.1 What is a Graph? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
11.2 Cycles Within Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315
11.3 Planar Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322
11.4 Counting Labeled Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327
Contents xi
11.5 Pólya Theory Revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331
Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359