PREFACE ix

**PART I THE BASICS OF ENUMERATIVE COMBINATORICS**

**1 Initial EnCOUNTers with Combinatorial Reasoning 3**

1.1 Introduction, 3

1.2 The Pigeonhole Principle, 3

1.3 Tiling Chessboards with Dominoes, 13

1.4 Figurate Numbers, 18

1.5 Counting Tilings of Rectangles, 24

1.6 Addition and Multiplication Principles, 33

1.7 Summary and Additional Problems, 46

References, 50

**2 Selections, Arrangements, and Distributions 51**

2.1 Introduction, 51

2.2 Permutations and Combinations, 52

2.3 Combinatorial Models, 64

2.4 Permutations and Combinations with Repetitions, 77

2.5 Distributions to Distinct Recipients, 86

2.6 Circular Permutations and Derangements, 100

2.7 Summary and Additional Problems, 109

Reference, 112

**3 Binomial Series and Generating Functions 113**

3.1 Introduction, 113

3.2 The Binomial and Multinomial Theorems, 114

3.3 Newton’s Binomial Series, 122

3.4 Ordinary Generating Functions, 131

3.5 Exponential Generating Functions, 147

3.6 Summary and Additional Problems, 163

References, 166

**4 Alternating Sums, Inclusion-Exclusion Principle, Rook Polynomials, and Fibonacci Nim 167**

4.1 Introduction, 167

4.2 Evaluating Alternating Sums with the DIE Method, 168

4.3 The Principle of Inclusion–Exclusion (PIE), 179

4.4 Rook Polynomials, 191

4.5 (Optional) Zeckendorf Representations and Fibonacci Nim, 202

4.6 Summary and Additional Problems, 207

References, 210

**5 Recurrence Relations 211**

5.1 Introduction, 211

5.2 The Fibonacci Recurrence Relation, 212

5.3 Second-Order Recurrence Relations, 222

5.4 Higher-Order Linear Homogeneous Recurrence Relations, 233

5.5 Nonhomogeneous Recurrence Relations, 247

5.6 Recurrence Relations and Generating Functions, 257

5.7 Summary and Additional Problems, 268

References, 273

**6 Special Numbers 275**

6.1 Introduction, 275

6.2 Stirling Numbers, 275

6.3 Harmonic Numbers, 296

6.4 Bernoulli Numbers, 306

6.5 Eulerian Numbers, 315

6.6 Partition Numbers, 323

6.7 Catalan Numbers, 335

6.8 Summary and Additional Problems, 345

References, 352

**PART II TWO ADDITIONAL TOPICS IN ENUMERATION**

**7 Linear Spaces and Recurrence Sequences 355**

7.1 Introduction, 355

7.2 Vector Spaces of Sequences, 356

7.3 Nonhomogeneous Recurrences and Systems of Recurrences, 367

7.4 Identities for Recurrence Sequences, 378

7.5 Summary and Additional Problems, 390

**8 Counting with Symmetries 393**

8.1 Introduction, 393

8.2 Algebraic Discoveries, 394

8.3 Burnside’s Lemma, 407

8.4 The Cycle Index and P´olya’s Method of Enumeration, 417

8.5 Summary and Additional Problems, 430

References, 432

**PART III NOTATIONS INDEX, APPENDICES, AND SOLUTIONS TO SELECTED ODD PROBLEMS**

Index of Notations 435

**Appendix A Mathematical Induction 439**

A.1 Principle of Mathematical Induction, 439

A.2 Principle of Strong Induction, 441

A.3 Well Ordering Principle, 442

**Appendix B Searching the Online Encyclopedia of Integer Sequences (OEIS) 443**

B.1 Searching a Sequence, 443

B.2 Searching an Array, 444

B.3 Other Searches, 444

B.4 Beginnings of OEIS, 444

**Appendix C Generalized Vandermonde Determinants 445**

Hints, Short Answers, and Complete Solutions to Selected Odd Problems 449

INDEX 467