Introduction to Combinatorics Book

Preface xi

1 Basic Counting Methods 1

1.1 The multiplication principle 1

1.2 Permutations 4

1.3 Combinations 6

1.4 Binomial coefficient identities 10

1.5 Distributions 19

1.6 The principle of inclusion and exclusion 23

1.7 Fibonacci numbers 31

1.8 Linear recurrence relations 33

1.9 Special recurrence relations 41

1.10 Counting and number theory 45

Notes 50

2 Generating Functions 53

2.1 Rational generating functions 53

2.2 Special generating functions 63

2.3 Partition numbers 76

2.4 Labeled and unlabeled sets 80

2.5 Counting with symmetry 86

2.6 Cycle indexes 93

2.7 Pólya’s theorem 96

2.8 The number of graphs 98

2.9 Symmetries in domain and range 102

2.10 Asymmetric graphs 103

Notes 105

3 The Pigeonhole Principle 107

3.1 Simple examples 107

3.2 Lattice points, the Gitterpunktproblem, and SET® 110

3.3 Graphs 115

3.4 Colorings of the plane 118

3.5 Sequences and partial orders 119

3.6 Subsets 124

Notes 126

4 Ramsey Theory 131

4.1 Ramsey’s theorem 131

4.2 Generalizations of Ramsey’s theorem 135

4.3 Ramsey numbers, bounds, and asymptotics 139

4.4 The probabilistic method 143

4.5 Sums 145

4.6 Van der Waerden’s theorem 146

Notes 150

5 Codes 153

5.1 Binary codes 153

5.2 Perfect codes 156

5.3 Hamming codes 158

5.4 The Fano Configuration 162

Notes 168

6 Designs 171

6.1 t-designs 171

CONTENTS ix

6.2 Block designs 175

6.3 Projective planes 180

6.4 Latin squares 182

6.5 MOLS and OODs 185

6.6 Hadamard matrices 188

6.7 The Golay code and S(5, 8, 24) 194

6.8 Lattices and sphere packings 197

6.9 Leech’s lattice 199

Notes 201

A Web Resources 205

B Notation 207

Exercise Solutions 211

References 225

Index 227