Combinatorics: A Problem-Based Approach (Problem Books in Mathematics)

372 Pages·2019·2.834 MB·English

by  Pavle Mladenović

Problem Books in Mathematics
Pavle Mladenović
Combinatorics
A Problem-Based Approach
Problem Books in Mathematics
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DepartmentofMathematics
DartmouthCollege
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USA
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Pavle Mladenovic´
Combinatorics
A Problem-Based Approach
123
PavleMladenovic´
FacultyofMathematics
UniversityofBelgrade
Belgrade,Serbia
ISSN0941-3502 ISSN2197-8506 (electronic)
ProblemBooksinMathematics
ISBN978-3-030-00830-7 ISBN978-3-030-00831-4 (eBook)
https://doi.org/10.1007/978-3-030-00831-4
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Preface
Thisbookisanextendedversionofthelatesteditionofmybookwritten
in Serbian under the title Combinatorics. I started gathering and selecting
material and combinatorial problems for the book in the late 1980s while
givinglecturestoyoungtalentedmathematicianspreparingfornationaland
international mathematical competitions and while working as a jury mem-
ber at the national mathematical olympiad.
In the 1990s and at the beginning of this century, till 2006, I was also
involved in organizing national mathematical competitions. In two five-year
periods,IwasleaderoftheteamofFRYugoslavia(SerbiaandMontenegro),
first at the Balkan Mathematical Olympiad (BMO1992–1996) and then at
the International Mathematical Olympiad (IMO1997–2001) and also was a
jurymemberatthesemathematicalolympiads. Thisengagementalongwith
myworkattheUniversityofBelgradehadagreatinfluenceontheformand
content of this book.
The first Serbian edition was published in 1989, with two subsequent
editions and one more printing in 1992, 2001, and 2013.
In Chapters 2–4, we introduce the basic combinatorial configurations
thatappearasasolutiontomanycombinatorialproblems,thebinomialand
multinomial theorems, and the inclusion-exclusion principle. The content of
these chapters is standard in every textbook in enumerative combinatorics.
The sources that were partially used here are the following: [5, 13, 15–
17, 23, 27].
Chapter 5 introduces the notion of generating functions of a sequence
ofrealnumbers,whichisapowerfultoolinenumerativecombinatorics. The
sources that were used for this chapter are [4, 16, 19, 24, 27]. Chapter 6 is
devoted to partitions of finite sets and partitions of positive integers. For
moredetails,see[1,5]. Chapter7dealswithBurnside’sLemmathatgivesa
methodofcountingequivalenceclassesdeterminedbyanequivalencerelation
V
VI Preface
on a finite set. For this topic and the more general Pólya enumeration
theorem, see [26, 27, 30].
InChapters8and9,weintroducethebasicnotionsandresultsofgraph
theory. The reader can find more details in [6, 14, 18, 20, 21, 30].
Several topics related to the existence of combinatorial configurations
are presented in Chapter 10, and the sources used are books [2, 5] and
journals [33, 34]. Several mathematical games are given in Chapter 11.
Topics from elementary probability are presented in Chapter 12. The
interested reader can find a detailed presentation of discrete probabilistic
models in Volume I of Feller’s book [3].
Inadditiontoelementaryexercises,allthechaptersalsocontainseveral
problems of medium difficulty and some challenging problems often selected
from materials that were suggested at national mathematical competitions
in different countries. Chapter 13 contains challenging problems that are
classified in seven sections. Several combinatorial problems that were sug-
gested at BMO and IMO are included here. The sources that were used
for theexercisesandmore challenging problems arethefollowing books and
journals: [7–12, 18, 22, 27–34].
Solutions to the exercises and problems from all the chapters are given
at the end of the book. For a small number of problems that are easier or
similar to previous problems, only answers or hints are provided.
I believe that the book may be useful not only to young talented math-
ematiciansinterestedinmathematicalolympiadsandtheirteachersbutalso
to researchers who apply combinatorial methods in different areas.
I would like to thank the anonymous referees for their useful comments
and suggestion to include topics related to graph theory and elementary
probability. I thank many colleagues from the Mathematical Society of
Serbia and the members of the juries of BMO and IMO for sharing their
experience and literature related to mathematical olympiads.
Finally, I am also very grateful to Mrs Alice Coople-Tošić for her lan-
guage assistance during the final preparation of the book.
Belgrade, Serbia Pavle Mladenović
2018
Contents
Preface V
1 Introduction 1
1.1 Sets, Functions, and Relations. . . . . . . . . . . . . . . . . . 1
1.2 Basic Combinatorial Rules . . . . . . . . . . . . . . . . . . . . 4
1.3 On the Subject of Combinatorics . . . . . . . . . . . . . . . . 6
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Arrangements, Permutations, and Combinations 9
2.1 Arrangements . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Arrangements Without Repetitions . . . . . . . . . . . . . . . 10
2.3 Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.4 Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.5 Arrangements of a Given Type . . . . . . . . . . . . . . . . . 13
2.6 Combinations with Repetitions Allowed . . . . . . . . . . . . 15
2.7 Some More Examples . . . . . . . . . . . . . . . . . . . . . . 16
2.8 A Geometric Method of Counting Arrangements . . . . . . . 21
2.9 Combinatorial Identities . . . . . . . . . . . . . . . . . . . . . 24
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3 Binomial and Multinomial Theorems 35
3.1 The Binomial Theorem . . . . . . . . . . . . . . . . . . . . . 35
3.2 Properties of Binomial Coefficients . . . . . . . . . . . . . . . 37
3.3 The Multinomial Theorem . . . . . . . . . . . . . . . . . . . . 42
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4 Inclusion-Exclusion Principle 49
4.1 The Basic Formula . . . . . . . . . . . . . . . . . . . . . . . . 49
4.2 The Special Case . . . . . . . . . . . . . . . . . . . . . . . . . 51
VII
VIII Contents
4.3 Some More Examples . . . . . . . . . . . . . . . . . . . . . . 52
4.4 Generalized Inclusion-Exclusion Principle . . . . . . . . . . . 58
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5 Generating Functions 63
5.1 Definition and Examples . . . . . . . . . . . . . . . . . . . . . 63
5.2 Operations with Generating Functions . . . . . . . . . . . . . 65
5.3 The Fibonacci Sequence . . . . . . . . . . . . . . . . . . . . . 67
5.4 The Recursive Equations. . . . . . . . . . . . . . . . . . . . . 69
5.5 The Catalan Numbers . . . . . . . . . . . . . . . . . . . . . . 70
5.6 Exponential Generating Functions . . . . . . . . . . . . . . . 72
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
6 Partitions 75
6.1 Partitions of Positive Integers . . . . . . . . . . . . . . . . . . 75
6.2 Ordered Partitions of Positive Integers . . . . . . . . . . . . . 78
6.3 Graphical Representation of Partitions . . . . . . . . . . . . . 80
6.4 Partitions of Sets . . . . . . . . . . . . . . . . . . . . . . . . . 84
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
7 Burnside’s Lemma 91
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
7.2 On Permutations . . . . . . . . . . . . . . . . . . . . . . . . . 92
7.3 Orbits and Cycles . . . . . . . . . . . . . . . . . . . . . . . . 95
7.4 Permutation Groups . . . . . . . . . . . . . . . . . . . . . . . 99
7.5 Burnside’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . 102
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
8 Graph Theory: Part 1 107
8.1 The Königsberg Bridge Problem . . . . . . . . . . . . . . . . 107
8.2 Basic Notions . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
8.3 Complement Graphs and Subgraphs . . . . . . . . . . . . . . 113
8.4 Paths and Connected Graphs . . . . . . . . . . . . . . . . . . 115
8.5 Isomorphic Graphs . . . . . . . . . . . . . . . . . . . . . . . . 117
8.6 Euler’s Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . 119
8.7 Hamiltonian Graphs . . . . . . . . . . . . . . . . . . . . . . . 120
8.8 Regular Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . 121
8.9 Bipartite Graphs . . . . . . . . . . . . . . . . . . . . . . . . . 122
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
9 Graph Theory: Part 2 127
9.1 Trees and Forests . . . . . . . . . . . . . . . . . . . . . . . . . 127
9.2 Planar Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . 130
Contents IX
9.3 Euler’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 133
9.4 Dual Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
9.5 Graph Coloring . . . . . . . . . . . . . . . . . . . . . . . . . . 137
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
10 Existence of Combinatorial Configurations 141
10.1 Magic Squares . . . . . . . . . . . . . . . . . . . . . . . . . . 141
10.2 Latin Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
10.3 System of Distinct Representatives . . . . . . . . . . . . . . . 148
10.4 The Pigeonhole Principle . . . . . . . . . . . . . . . . . . . . 152
10.5 Ramsey’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . 153
10.6 Arrow’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 156
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
11 Mathematical Games 165
11.1 The Nim Game . . . . . . . . . . . . . . . . . . . . . . . . . . 165
11.2 Golden Ratio in a Mathematical Game . . . . . . . . . . . . . 167
11.3 Game of Fifteen . . . . . . . . . . . . . . . . . . . . . . . . . 168
11.4 Conway’s Game of Reaching a Level . . . . . . . . . . . . . . 170
11.5 Two More Games . . . . . . . . . . . . . . . . . . . . . . . . . 174
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
12 Elementary Probability 177
12.1 Discrete Probability Space . . . . . . . . . . . . . . . . . . . . 177
12.2 Conditional Probability and Independence . . . . . . . . . . . 183
12.3 Discrete Random Variables . . . . . . . . . . . . . . . . . . . 185
12.4 Mathematical Expectation . . . . . . . . . . . . . . . . . . . . 187
12.5 Law of Large Numbers . . . . . . . . . . . . . . . . . . . . . . 191
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
13 Additional Problems 199
13.1 Basic Combinatorial Configurations. . . . . . . . . . . . . . . 199
13.2 Square Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
13.3 Combinatorics on a Chessboard . . . . . . . . . . . . . . . . . 203
13.4 The Counterfeit Coin Problem . . . . . . . . . . . . . . . . . 205
13.5 Extremal Problems on Finite Sets . . . . . . . . . . . . . . . 206
13.6 Combinatorics at Mathematical Olympiads . . . . . . . . . . 209
13.7 Elementary Probability . . . . . . . . . . . . . . . . . . . . . 217
14 Solutions 221
14.1 Solutions for Chapter 1 . . . . . . . . . . . . . . . . . . . . . 221
14.2 Solutions for Chapter 2 . . . . . . . . . . . . . . . . . . . . . 221
14.3 Solutions for Chapter 3 . . . . . . . . . . . . . . . . . . . . . 235