Introduction to Combinatorics

by Erickson, Martin J.

• ISBN13: 9781118637531
• ISBN10: 1118637534
• Edition: 2nd
• Format: Hardcover
• Copyright: 2013-04-08
• Publisher: Wiley

Summary

This book provides an introduction to the main concepts of combinatorics, features fundamental results, discusses interconnection and problem-solving techniques, and collects and disseminates open problems that raise questions and observations.is the ideal text for advanced undergraduate and early graduate courses in this subject. The Second Edition contains over fifty new examples that illustrate important combinatorial concepts and range from the routine (i.e. special kinds of sets, functions, and sequences) to the advanced (i.e. the SET game, the Gitterpunktproblem, and enumeration of partial orders). The tables and references are been updated throughout, reflecting advances in Ramsey numbers and Thomas Hales' solution of Kepler's conjecture). In addition, many exciting new computer programs and exercises have been incorporated to help readers understand and apply combinatorial techniques and ideas. The author has now made it possible for readers to encode and execute programs for formulas that were previously inaccessible, allowing for a deeper, investigative study of combinatorics. Each of the book's three sections, Existence, Enumeration, and Construction, begin with a simply stated first principle, which is then developed step-by-step until it leads to one of the three major achievements of combinatorics: Van der Waerden's theorem on arithmetic progressions, Polya's graph enumeration formula, and Leech's 24-dimensional lattice. Many important combinatorial methods are revisited and repeated several times throughout the book in exercises, examples, theorems, and proofs alike, enabling readers to build confidence and reinforce their understanding of complex material.

Author Biography

MARTIN J. ERICKSON, PhD, is Professor in the Department of Mathematics at Truman State University. The author of numerous books, including Mathematics for the Liberal Arts (Wiley), he is a member of the American Mathematical Society, Mathematical Association of America, and American Association of University Professors.

Table of Contents

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