Duane DeTemple & William Webb 
Combinatorial Reasoning 
An Introduction to the Art of Counting

Written by two well-known scholars in the field,
Combinatorial Reasoning: An Introduction to the Art of
Counting presents a clear and comprehensive introduction to the
concepts and methodology of beginning combinatorics. Focusing on
modern techniques and applications, the book develops a variety of
effective approaches to solving counting problems.

Balancing abstract ideas with specific topical coverage, the
book utilizes real world examples with problems ranging from basic
calculations that are designed to develop fundamental concepts to
more challenging exercises that allow for a deeper exploration of
complex combinatorial situations. Simple cases are treated first
before moving on to general and more advanced cases. Additional
features of the book include:

* Approximately 700 carefully structured problems designed
for readers at multiple levels, many with hints and/or short
answers

* Numerous examples that illustrate problem solving using
both combinatorial reasoning and sophisticated algorithmic
methods

* A novel approach to the study of recurrence sequences,
which simplifies many proofs and calculations

* Concrete examples and diagrams interspersed throughout to
further aid comprehension of abstract concepts

* A chapter-by-chapter review to clarify the most crucial
concepts covered

Combinatorial Reasoning: An Introduction to the Art of
Counting is an excellent textbook for upper-undergraduate and
beginning graduate-level courses on introductory combinatorics and
discrete mathematics.

Table of Content

PREFACE ix

PART I THE BASICS OF ENUMERATIVE COMBINATORICS

1 Initial En COUNTers 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 Pólya’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

About the author

DUANE DETEMPLE, PHD, is
Professor Emeritus in the Department of Mathematics at Washington
State University (WSU). He is the recipient of the 2007 WSU Sahlin
Faculty Excellence Award for Instruction as well as the
Distinguished Teaching Award from the Pacific Northwest Section of
the Mathematical Association of America.

WILLIAM WEBB, PHD, is Professor in the
Department of Mathematics at Washington State University and
President of the Fibonacci Association. His research interests
include the properties of recurrence sequences and binomial
coefficients. He is the author of numerous research publications on
combinatorics, number theory, fair division, and cryptography.