Proofs that Really Count: The Art of Combinatorial Proof (Dolciani Mathematical Expositions)

In making the gift, Professor Dolciani, herself an exceptionally talented and successful expositor of mathematics, had the purpose of furthering the ideal of excellence in mathematical exposition. The Association, for its part, was delighted to accept the gracious gesture initiating the revolving fund for this series from one who has served the Association with distinction, both as a member of the Committee on Publications and as a member of the Board of Governors. It was with genuine pleasure that the Board chose to name the series in her honor. The books in the series are selected for their lucid expository style and stimulating mathematical content. Typically, they contain an ample supply of exercises, many with accompanying solutions. They are intended to be sufficiently elementary for the undergraduate and even the mathematically inclined high-school student to understand and enjoy, but also to be interesting and sometimes challenging to the more advanced mathematician. ## Who counts? We are pleased to acknowledge the many people who made this book possible|either directly or indirectly. Those who came before us are responsible for the rise in popularity of combinatorial proof. Books whose importance cannot be overlooked are Constructive Combinatorics by Dennis Stanton and Dennis White, Enumerative Combinatorics Volumes 1 & 2 by Richard Stanley, Combinatorial Enumeration by Ian Goulden and David Jackson, and Concrete Mathematics by Ron Graham, Don Knuth & Oren Patashnik. In addition to these mathematicians, others whose works continue to inspire us include Mathematics is the science of patterns, and mathematicians attempt to understand these patterns and discover new ones using a variety of tools. In Proofs That Really Count, award-winning math professors Arthur Benjamin and Jennifer Quinn demonstrate that many number patterns, even very complex ones, can be understood by simple counting arguments. The arguments primarily take one of two forms: - A counting question is posed and answered in two different ways. Since both answers solve the same question they must be equal. -Two different sets are described, counted, and a correspondence found between them. One-to-one correspondences guarantee sets of the same size. Almost one-to-one correspondences take error terms into account. Even many-to-one correspondences are utilized. The book explores more than 200 identities throughout the text and exercises, frequently emphasizing numbers not often thought of as numbers that count: Fibonacci Numbers, Lucas Numbers, Continued Fractions, and Harmonic Numbers, to name a few. Numerous hints and references are given for all chapter exercises and many chapters end with a list of identities in need of combinatorial proof. The extensive appendix of identities will be a valuable resource. This book should appeal to readers of all levels, from high school math students to professional mathematicians "Mathematics is the science of patterns, and mathematicians attempt to understand these patterns and discover new ones using a variety of tools. In Proofs That Really Count, award-winning math professors Arthur Benjamin and Jennifer Quinn demonstrate that many number patterns, even very complex ones, can be understood by simple counting arguments. The book emphasizes numbers that are often not thought of as numbers that count: Fibonacci Numbers, Lucas Numbers, Continued Fractions, and Harmonic Numbers, to name a few. Numerous hints and references are given for all chapter exercises and many chapters end with a list of identities in need of combinatorial proof. The extensive appendix of identities will be a valuable resource. This book should appeal to readers of all levels, from high school math students to professional mathematicians."--Publisher's description Cover Title Foreword Contents 1. Fibonacci Identities 2. Gibonacci and Lucas Identities 3. Linear Recurrences 4. Continued Fractions 5. Binomial Identities 6. Alternating Sign Binomial Identities 7. Harmonic and Stirling Number Identities 8. Number Theory 9. Advanced Fibonacci & Lucas Identities Some Hints and Solutions for Chapter Exercises Appendix of Combinatorial Theorems Appendix of Identities Bibliography Index Back Cover Arthur T. Benjamin And Jennifer J. Quinn. Includes Bibliographical References (p. 187-190) And Index.