Combinatorics: A Guided Tour

by David R. Mazur

Combinatorics is mathematics of enumeration, existence, construction, and optimization questions concerning finite sets. This text focuses on the first three types of questions and covers basic counting and existence principles, distributions, generating functions, recurrence relations, Polya theory, combinatorial designs, error correcting codes, partially ordered sets, and selected applications to graph theory including the enumeration of trees, the chromatic polynomial, and introductory Ramsey theory. The only prerequisites are single-variable calculus and familiarity with sets and basic proof techniques. The text emphasizes the brands of thinking that are characteristic of bijective and combinatorial proofs, recursive analysis, and counting problem classification. It is flexible enough to be used for undergraduate courses in combinatorics, second courses in discrete mathematics, introductory graduate courses in applied mathematics programs, as well as for independent study or reading courses. What makes this text a guided tour are the approximately 350 reading questions spread throughout its eight chapters. These questions provide checkpoints for learning and prepare the reader for the end-of-section exercises of which there are over 470. Most sections conclude with Travel Notes that add color to the material of the section via anecdotes, open problems, suggestions for further reading, and biographical information about mathematicians involved in the discoveries.

Genres: Mathematics

410 pages, Hardcover

First published May 1, 2010

Reviews

[Charles · Rating 5 out of 5]

Combinatorics is an area of mathematics that even novices can find fun. Questions involving counting socks, the number of hairs on the heads of people and if two people in a group have the same birthday are all easy to understand and evaluate. When I teach finite math, combinatorics is regularly rated the topic most easy to understand.
However, this ease of understanding does not mean that combinatorics is not important, it has applications in many areas and additional applications will surely be discovered. Combinatorics is an excellent candidate for a special topics course for mathematics majors; with the broad spectrum of applications the course can simultaneously be an advanced and a capstone course.
This book would be an excellent selection for the textbook of such a course. The explanations are at an appropriate level for the audience and there are exercises at the end of each section. Solutions to some of them are included in an appendix. The coverage is also of sufficient breadth; all of the major areas of combinatorics are covered. Calculus is not a prerequisite; the most complex task is understanding the notation, which can at times be difficult as it is compact. This is normal; mathematics uses a very condensed language with a few symbols representing a sequence of operations, some of which may be abstract.
This book is the best candidate for a textbook in combinatorics that I have encountered.

Published in Journal of Recreational Mathematics, reprinted with permission

[Alex · Rating 5 out of 5]

Especially useful were the mini-exercises embedded within the text of each chapter. This would be a good book for self-study.