Combinatorial Reasoning

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 Polya’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