A Walk Through Combinatorics

The first half of the book walks the reader through methods of counting, both direct elementary methods and the more advanced method of generating functions. Then, in the second half of the book, the reader learns how to apply these methods to fascinating objects, such as graphs, designs, random variables, partially ordered sets, and algorithms. In short, the first half emphasizes depth by discussing counting methods at length; the second half aims for breadth, by showing how numerous the applications of our methods are.

New to this fifth edition of A Walk Through Combinatorics is the addition of Instant Check exercises — more than a hundred in total — which are located at the end of most subsections. As was the case for all previous editions, the exercises sometimes contain new material that was not discussed in the text, allowing instructors to spend more time on a given topic if they wish to do so. With a thorough introduction into enumeration and graph theory, as well as a chapter on permutation patterns (not often covered in other textbooks), this book is well suited for any undergraduate introductory combinatorics class.

Readership: This book is primarily suitable for advanced undergraduate students and instructors teaching an introductory combinatorics class. However, it would also be applicable to graduate students in fields other than combinatorics who need an introduction to Combinatorics.

Table of Contents

I. Basic Methods

1. Seven Is More Than Six. The Pigeon-Hole Principle

2. One Step at a Time. The Method of Mathematical Induction

II. Enumerative Combinatorics

3. There are a Lot of Them. Elementary Counting Problems

4. No Matter How You Slice It. The Binomial Theorem and Related Identities

5. Divide and Conquer. Partitions

6. Not So Vicious Cycles. Cycles in Permutations

7. You Shall Not Overcount. The Sieve

8. A Function is Worth Many Numbers. Generating Functions

III. Graph Theory

9. Dots and Lines. The Origins of Graph Theory

10. Staying Connected. Trees

11. Finding A Good Match. Coloring and Matching

12. Do Not Cross. Planar Graphs

V. Horizons

13. Does it Clique? Ramsey Theory

14. So Hard To Avoid. Subsequence Conditions on Permutations

15. Who Knows What it Looks Like, But it Exists. The Probabilistic Method

16. At Least Some Order. Partial Orders and Lattices

17. As Evenly As Possible. Block Designs and Error Correcting Codes

18. Are They Really Different? Counting Unlabeled Structures

19. The Sooner The Better. Combinatorial Algorithms

20. Does Many Mean More Than One? Computational Complexity