Titel: A Path to Combinatorics for Undergraduates
Autor/en: Titu Andreescu, Zuming Feng
11. November 2003 - kartoniert - 228 Seiten
This unique approach to combinatorics is centered around challenging examples, fully-worked solutions, and hundreds of problems---many from Olympiads and other competitions, and many original to the authors. Each chapter highlights a particular aspect of the subject and casts combinatorial concepts in the guise of questions, illustrations, and exercises that are designed to encourage creativity, improve problem-solving techniques, and widen the reader's mathematical horizons.
Topics encompass permutations and combinations, binomial coefficients and their applications, recursion, bijections, inclusions and exclusions, and generating functions. The work is replete with a broad range of useful methods and results, such as Sperner's Theorem, Catalan paths, integer partitions and Young's diagrams, and Lucas' and Kummer's Theorems on divisibility. Strong emphasis is placed on connections between combinatorial and graph-theoretic reasoning and on links between algebra and geometry.
The authors' previous text, 102 Combinatorial Problems, makes a fine companion volume to the present work, which is ideal for Olympiad participants and coaches, advanced high school students, undergraduates, and college instructors. The book's unusual problems and examples will stimulate seasoned mathematicians as well. A Path to Combinatorics for Undergraduates is a lively introduction not only to combinatorics, but also to mathematical ingenuity, rigor, and the joy of solving puzzles.
Preface * Introduction * Acknowledgments * Abbreviations and Notations * Addition on Multiplication? * Combinations * Properties of Binomial Coefficients * Bijections * Inclusions and Exclusions * Recursions * Calculating in Two Ways --- Fubini's Principle * Generating Functions * Review Exercises * Glossary * Further Reading
From the reviews:
"A good foundation in combinatorics is provided in the early chapters that cover ideas in combinatorial geometry.... This book serves as a solid stepping stone for more advanced combinatorics studies in related mathematical science fields or in computer science."
- L'Enseignement Mathématique
"This book is an introduction to counting strategies in combinatorial theory. The main mathematical ideas are carefully worked into organized, challenging, and instructive examples given in the nine chapters of this book. In the last chapter we find 111 problems (without solutions). The greater part of them are from various mathematical contests. The...experience of the authors in preparing students for various mathematical competitions allowed them to present a big collection of beautiful problems. By studying this book, undergraduates will be well-equipped to further their knowledge in more abstract combinatorics and its related fields."
"...the book provides quite an amazing collection of combinatorial problems, many of them original, and many of them from a hard to find sources like Russian olympiads. (...) The presentation of the solutions is very clear and instructive, with emphasis on common mistakes."
"The goal of the book is to explain the main concepts and ideas of combinatorics to undergraduate students. Extremely helpful is the extensive use of examples for explanation purposes, which makes this book so pleasant to read. All the ideas and problems are addressed by giving rich examples, and what is even more, each example is solved in high detail immediately after it is posed. ... So it is highly recommended to read everything in the book." (Simon Seichter, Simulation News Europe, Vol. 16 (1), 2006)