How to Count

1. Front Matter

   Pages i-xv

   PDF

2. Preliminaries

   • Robert A. Beeler

   Pages 1-20

3. Basic Counting

   • Robert A. Beeler

   Pages 21-57

4. The Binomial Coefficient

   • Robert A. Beeler

   Pages 59-94

5. Distribution Problems

   • Robert A. Beeler

   Pages 95-114

6. Generating Functions

   • Robert A. Beeler

   Pages 115-146

7. Recurrence Relations

   • Robert A. Beeler

   Pages 147-194

8. Advanced Counting—Inclusion and Exclusion

   • Robert A. Beeler

   Pages 195-218

9. Advanced Counting—Pólya Theory

   • Robert A. Beeler

   Pages 219-255

10. Application: Probability

    • Robert A. Beeler

    Pages 257-289

11. Application: Combinatorial Designs

    • Robert A. Beeler

    Pages 291-307

12. Application: Graph Theory

    • Robert A. Beeler

    Pages 309-343

13. Back Matter

    Pages 345-361

    PDF

Providing a self-contained resource for upper undergraduate courses in combinatorics, this text emphasizes computation, problem solving, and proof technique. In particular, the book places special emphasis the Principle of Inclusion and Exclusion and the Multiplication Principle. To this end, exercise sets are included at the end of every section, ranging from simple computations (evaluate a formula for a given set of values) to more advanced proofs. The exercises are designed to test students' understanding of new material, while reinforcing a working mastery of the key concepts previously developed in the book. Intuitive descriptions for many abstract techniques are included. Students often struggle with certain topics, such as generating functions, and this intuitive approach to the problem is helpful in their understanding. When possible, the book introduces concepts using combinatorial methods (as opposed to induction or algebra) to prove identities. Students are also asked to prove identities using combinatorial methods as part of their exercises. These methods have several advantages over induction or algebra.

• Polya theory
• applied combinatorics
• discrete mathematics
• enumerative combinatorics
• probability
• combinatorics

“The book is an excellent introduction to combinatorics. … The author uses a clear language and often provides an easy intuitive access to abstract subjects. The presentation is well motivated, the explanations are transparent and illustrated by carefully selected examples. Each section ends with a list of well formulated exercises which make the book ideally suited for self-instruction.” (Astrid Reifegerste, zbMATH 1328.05001, 2016)

“This book by Beeler … is an excellent introductory text on combinatorics. The author gives the right balance of theory, computation, and applications, and he presents introductory-level topics, such as the multiplication principle, binomial theorem, and distribution problems in a clear manner. … Summing Up: Highly recommended. Upper-division undergraduates through researchers and faculty.” (S. L. Sullivan, Choice, Vol. 53 (1), September, 2015)

• Department of Mathematics and Statistics, East Tennessee State University, Johnson City, USA

  Robert A. Beeler

Robert A. Beeler is an Associate Professor of Mathematics at East Tennessee State University. His research interests include enumerative combinatorics and graph theory; edge decompositions of graphs, graceful labelings on graphs, intersection/representation theory, and combinatorial designs; combinatorial games and games on graphs. He is a member of the Mathematics Association of America, the American Mathematical Society, and the Institute of Combinatorics and its Applications.

Policies and ethics