Combinatorics

Preface ix

Chapter 1 The Mathematics of Choice 1

1.1. The Fundamental Counting Principle 2

1.2. Pascal’s Triangle 10

* 1.3. Elementary Probability 21

* 1.4. Error-Correcting Codes 33

1.5. Combinatorial Identities 43

1.6. Four Ways to Choose 56

1.7. The Binomial and Multinomial Theorems 66

1.8. Partitions 76

1.9. Elementary Symmetric Functions 87

* 1.10. Combinatorial Algorithms 100

Chapter 2 The Combinatorics of Finite Functions 117

2.1. Stirling Numbers of the Second Kind 117

2.2. Bells, Balls, and Urns 128

2.3. The Principle of Inclusion and Exclusion 140

2.4. Disjoint Cycles 152

2.5. Stirling Numbers of the First Kind 161

Chapter 3 Polya’s Theory of Enumeration 175

3.1. Function Composition 175

3.2. Permutation Groups 184

3.3. Burnside’s Lemma 194

3.4. Symmetry Groups 206

3.5. Color Patterns 218

3.6. Polya’s Theorem 228

3.7. The Cycle Index Polynomial 241

Chapter 4 Generating Functions 253

4.1. Difference Sequences 253

4.2. Ordinary Generating Functions 268

4.3. Applications of Generating Functions 284

4.4. Exponential Generating Functions 301

4.5. Recursive Techniques 320

Chapter 5 Enumeration in Graphs 337

5.1. The Pigeonhole Principle 338

* 5.2. Edge Colorings and Ramsey Theory 347

5.3. Chromatic Polynomials 357

* 5.4. Planar Graphs 372

5.5. Matching Polynomials 383

5.6. Oriented Graphs 394

5.7. Graphic Partitions 408

Chapter 6 Codes and Designs 421

6.1. Linear Codes 422

6.2. Decoding Algorithms 432

6.3. Latin Squares 447

6.4. Balanced Incomplete Block Designs 461

Appendix A1 Symmetric Polynomials 477

Appendix A2 Sorting Algorithms 485

Appendix A3 Matrix Theory 495

Bibliography 501

Hints and Answers to Selected Odd-Numbered Exercises 503

Index of Notation 541

Index 547