ISBN-10:
1118652185
ISBN-13:
9781118652183
Pub. Date:
05/05/2014
Publisher:
Wiley
Combinatorial Reasoning: An Introduction to the Art of Counting / Edition 1

Combinatorial Reasoning: An Introduction to the Art of Counting / Edition 1

by Duane DeTemple, William Webb

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Overview

Combinatorial Reasoning: An Introduction to the Art of Counting / Edition 1

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

Balancing abstract ideas with specific topical coverage, thebook utilizes real world examples with problems ranging from basiccalculations that are designed to develop fundamental concepts tomore challenging exercises that allow for a deeper exploration ofcomplex combinatorial situations. Simple cases are treated firstbefore moving on to general and more advanced cases. Additionalfeatures of the book include:

• Approximately 700 carefully structured problems designedfor readers at multiple levels, many with hints and/or shortanswers
• Numerous examples that illustrate problem solving usingboth combinatorial reasoning and sophisticated algorithmicmethods
• A novel approach to the study of recurrence sequences,which simplifies many proofs and calculations
• Concrete examples and diagrams interspersed throughout tofurther aid comprehension of abstract concepts
• A chapter-by-chapter review to clarify the most crucialconcepts covered

Combinatorial Reasoning: An Introduction to the Art ofCounting is an excellent textbook for upper-undergraduate andbeginning graduate-level courses on introductory combinatorics anddiscrete mathematics.

Product Details

ISBN-13: 9781118652183
Publisher: Wiley
Publication date: 05/05/2014
Pages: 488
Product dimensions: 6.20(w) x 9.30(h) x 1.20(d)

About the Author

DUANE DETEMPLE, PHD, isProfessor Emeritus in the Department of Mathematics at WashingtonState University (WSU). He is the recipient of the 2007 WSU SahlinFaculty Excellence Award for Instruction as well as theDistinguished Teaching Award from the Pacific Northwest Section ofthe Mathematical Association of America.

WILLIAM WEBB, PHD, is Professor in theDepartment of Mathematics at Washington State University andPresident of the Fibonacci Association. His research interestsinclude the properties of recurrence sequences and binomialcoefficients. He is the author of numerous research publications oncombinatorics, number theory, fair division, and cryptography.

Table of Contents

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 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

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