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Combinatorics: A Guided Tour (MAA Textbooks), by David R. Mazur
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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, Pólya 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 combinatorics: 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.
- Sales Rank: #249211 in Books
- Brand: Brand: Mathematical Association of America
- Published on: 2009-11-18
- Original language: English
- Number of items: 1
- Dimensions: 10.00" h x 1.02" w x 7.01" l, 1.95 pounds
- Binding: Hardcover
- 391 pages
Features
- Used Book in Good Condition
Review
This is a well-written, reader-friendly, and self-contained undergraduate course on combinatorics, focusing on enumeration. The book includes plenty of exercises, and about hal of them come with hints. --M. Bona, Choice Magazine
...Combinatorics is an excellent candidate for a special topics course for mathematics majors; with the broad spectrum of applications that 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...The coverage is also sufficient of breadth; all of the major areas of combinatorics are covered...This book is the best candidate for a textbook in combinatorics that I have encountered. --Charles Ashbacher
I have reviewed many books on discrete mathematics over the years; some have been quite good...However, the book reviewed here, Combinatorics: A Guided Tour by David R. Mazur (hereafter CAGT) is the best book ever written on undergraduate combinatorial theory--it deserves an award. I like everything about this book, starting with the selection of topics and the organization.
...The delineation of the topics is first rate--better than I have ever seen in any other book. Their presentation is generally thorough, with much of it in the form of worked problems. The book is very much designed as a textbook; there are plenty of problems at the end of each section. ... CAGT has both good breadth and great presentation; it is in fact a new standard in presentation for combinatorics, essential as a resource for any instructor, including those teaching out of a different text. For the student: If you are just starting to build a library in combinatorics, this should be your first book. --The UMAP Journal
About the Author
David Mazur is Associate Professor of Mathematics at Western New England College in Springfield, Massachusetts. He received his undergraduate degree in mathematics from the University of Delaware in 1993. He then began graduate study at The Johns Hopkins University where he earned a Master's in 1996 and a Ph.D. in 1999, focusing on operations research, integer programming, and polyhedral combinatorics. He was a 2000-2001 Project NExT fellow and continues to be active in the Mathematical Association of America.
Most helpful customer reviews
5 of 5 people found the following review helpful.
A great book for supplement or self-study if it isn't already your textbook!
By C. Pettis
This is one of the best math books I have read, and I've read quite a few. It's a great text book and a great book for self-learning as well. It doesn't assume too much about prior knowledge, and does a good job developing the tools and background needed to understand the concepts. The class I had did not use this book, but I got it from a friend who did use it in his class. By the end, I totally relied on this book as the other book wasn't that helpful.
What stands out:
- Painstaking effort to develop notations (like in inclusion/exclusion) that suggest what is going on logically.
- Connecting a range of techniques together into an easily referenced area (e.g., counting techniques for distinguished/identical balls into distinguished/identical bins brought together in one table that was sequentially filled out as the techniques were discussed).
- Detailed pictures that are not just referenced but leveraged to great success when describing what is going on with a combinatorial counting technique.
- Consistent references in the book to thing where concepts were first introduced, and forward to where statements will be proven.
The biggest criticism I have of math books is when they discuss a concept and spend an inordinate amount of time discussing things that are easy, but when it comes to the difficult nut of the new concepts, just blows through that part with very little exposition. This book does NOT make that mistake. The author has a good grasp of what the difficult parts of a new concept and takes care to discuss those parts with care, anticipating hangups that the student might have with the concept and talking them through it.
To make this not seem like a totally gushing review, I'll point out one style disagreement. The author has inline questions (well-set from the text) that he asks you to consider but does not answer in the text. The questions are usually pretty easy to think through, and the remainder of the text either doesn't depend on the student having the answer or, if it is needed, actually does provide the answer later. However, that can be distracting if you get hung on a question you can't answer. Overall, it doesn't detract from the book, and again, I'd say is a style difference.
1 of 1 people found the following review helpful.
Excellent Textbook
By Ramin M
I agree with Pettis' review. I'm on Chapter 2 and I must say that the author is doing a great job presenting such a tough subject. I was having a hell of a time with Herstein's Topics in Algebra last semester and quickly left it in favor of Raleigh's First Course in Abstract Algebra, which understood it's audience. David Mazur also understands his audience and explains the most important things in great detail. So far, I like this book very much. However, I too think that the questions throughout the book are kind of a pebble in my shoe. I want to answer them, but I don't have the time to mess with them. I wish there was a solution manual which answered every question and every problem in this book. I'd pay $100 for such a book. I like the subject and I want to get deeper by going through more problems. Without the solutions manual, I kind of tend to just do only the assigned problems, because as it is they are already very tough. I have Schaums' book too, which has a huge number of solved problems and with that I am making progress in the class. But don't worry, the questions throughout the book can be skipped. I think they are for those that have the time to get closer to the subject, like instructors or those that study on their own. Bottom line, David Mazur did an excellent job.
0 of 0 people found the following review helpful.
Combinatorics a Guided tour
By D. York
It is an assigned text for the course I'm taking. This meets my needs and expectations. A Clear introduction to the topic.
See all 7 customer reviews...
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