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Birkhäuser

A Path to Combinatorics for Undergraduates

Counting Strategies

  • Textbook
  • © 2004

Overview

Authors:
  1. Titu Andreescu

    1. American Mathematics Competitions, University of Nebraska, Lincoln, USA

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    You can also search for this author in PubMed   Google Scholar

  2. Zuming Feng

    1. Department of Mathematics, Phillips Exeter Academy, Exeter, USA

    View author publications

    You can also search for this author in PubMed   Google Scholar

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Table of contents (9 chapters)

  1. Front Matter

    Pages i-xix
    Download chapter PDF

  2. Addition or Multiplication?

    • Titu Andreescu, Zuming Feng
    Pages 1-23

  3. Combinations

    • Titu Andreescu, Zuming Feng
    Pages 25-41

  4. Properties of Binomial Coefficients

    • Titu Andreescu, Zuming Feng
    Pages 43-67

  5. Bijections

    • Titu Andreescu, Zuming Feng
    Pages 69-90

  6. Recursions

    • Titu Andreescu, Zuming Feng
    Pages 91-116

  7. Inclusion-Exclusion Principle

    • Titu Andreescu, Zuming Feng
    Pages 117-141

  8. Calculating in Two Ways: Fubini’s Principle

    • Titu Andreescu, Zuming Feng
    Pages 143-164

  9. Generating Functions

    • Titu Andreescu, Zuming Feng
    Pages 165-193

  10. Review Exercises

    • Titu Andreescu, Zuming Feng
    Pages 195-212

  11. Back Matter

    Pages 213-228
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Keywords

About this book

The main goal of the two authors is to help undergraduate students understand the concepts and ideas of combinatorics, an important realm of mathematics, and to enable them to ultimately achieve excellence in this field. This goal is accomplished by familiariz­ ing students with typical examples illustrating central mathematical facts, and by challenging students with a number of carefully selected problems. It is essential that the student works through the exercises in order to build a bridge between ordinary high school permutation and combination exercises and more sophisticated, intricate, and abstract concepts and problems in undergraduate combinatorics. The extensive discussions of the solutions are a key part of the learning process. The concepts are not stacked at the beginning of each section in a blue box, as in many undergraduate textbooks. Instead, the key mathematical ideas are carefully worked into organized, challenging, and instructive examples. The authors are proud of their strength, their collection of beautiful problems, which they have accumulated through years of work preparing students for the International Math­ ematics Olympiads and other competitions. A good foundation in combinatorics is provided in the first six chapters of this book. While most of the problems in the first six chapters are real counting problems, it is in chapters seven and eight where readers are introduced to essay-type proofs. This is the place to develop significant problem-solving experience, and to learn when and how to use available skills to complete the proofs.

Reviews

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."

—MAA Online

"...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."

—Mathematics Bohemica

"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)

Authors and Affiliations

  • American Mathematics Competitions, University of Nebraska, Lincoln, USA

    Titu Andreescu

  • Department of Mathematics, Phillips Exeter Academy, Exeter, USA

    Zuming Feng

Bibliographic Information

  • Book Title: A Path to Combinatorics for Undergraduates

  • Book Subtitle: Counting Strategies

  • Authors: Titu Andreescu, Zuming Feng

  • DOI: https://doi.org/10.1007/978-0-8176-8154-8

  • Publisher: Birkhäuser Boston, MA

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer Science+Business Media New York 2004

  • Softcover ISBN: 978-0-8176-4288-4Published: 11 November 2003

  • eBook ISBN: 978-0-8176-8154-8Published: 01 December 2013

  • Edition Number: 1

  • Number of Pages: XIX, 228

  • Number of Illustrations: 39 b/w illustrations

  • Topics: Combinatorics, Geometry, Convex and Discrete Geometry, Probability Theory and Stochastic Processes

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