Enumerative Combinatorics, Vol. 1 (Cambridge Studies in Advanced Mathematics) [Hardcover]

Richard P. Stanley (Author), Gian-Carlo Rota (Foreword)

Book Description

ISBN-10: 0521553091 | ISBN-13: 978-0521553094 | Publication Date: April 13, 1997

This book, the first of a two-volume basic introduction to enumerative combinatorics, concentrates on the theory and application of generating functions, a fundamental tool in enumerative combinatorics. Richard Stanley covers those parts of enumerative combinatorics with the greatest applications to other areas of mathematics. The four chapters are devoted to an accessible introduction to enumeration, sieve methods--including the Principle of Inclusion-Exclusion, partially ordered sets, and rational generating functions. A large number of exercises, almost all with solutions, augment the text and provide entry into many areas not covered directly. Graduate students and research mathematicians who wish to apply combinatorics to their work will find this an authoritative reference.

Editorial Reviews

Review

"...sure to become a standard as an introductory graduate text in combinatorics."
Bulletin of the AMS

"Stanley's book is very readable and a mine of information."
Journal of the LMS

"...will engage from start to finish the attention of any mathematician who will open it at page one."
Gian-Carlo Rota

"...an excellent and valuable book."
Mathematical Reviews

Book Description

Enumerative combinatorics deals with the basic problem of counting how many objects have a given property. Since this problem arises in many areas of mathematics and science the subject is of great applicability. This book provides an introduction to the subject at a level suitable for graduate students and research mathematicians. The author has made a special effort, by providing extensive exercises with solutions, to show the broad applicability of enumerative combinatorics to other areas of mathematics.

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Product Details

• Hardcover: 340 pages
• Publisher: Cambridge University Press (April 13, 1997)
• Language: English
• ISBN-10: 0521553091
• ISBN-13: 978-0521553094
• Product Dimensions: 9.2 x 6.3 x 0.9 inches
• Shipping Weight: 1.4 pounds (View shipping rates and policies)
• Average Customer Review: 5.0 out of 5 stars  See all reviews (6 customer reviews)
• Amazon Best Sellers Rank: #827,534 in Books (See Top 100 in Books)

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18 of 21 people found the following review helpful:

5.0 out of 5 stars This is for people who likes to COUNT, February 25, 2004

By 

bal gombak (Cambridge, MA USA) - See all my reviews

This review is from: Enumerative Combinatorics, Volume 2 (Paperback)

Gosh! This is for people who count, what else does a combinatorist do? Before people dismiss me as somebody who don't know hoot about math: I took a class with Prof. Stanley (the author) in college, and I had actually used vol 1 as a text. The material is highbrow (I agree on the 'hardcore' math observation) but the main theme of the book is how to 'count' -- needless to say not in the sense of everyday counting, but in the sense that 'topology' is 'coffee-to-donut transformation' and 'analysis' is 'honors calculus'. You have to know how to count, and comfortable with combinatorial proof to actually learn from this. I like the fact that Prof. Stanley asks for combinatorial proof to some known results, marking them as unsolved -- he really elevates the status of combinatorial proof, a method many dismiss as 'handwaving'. There is a number given to each exercise, according to the level of difficulty: [1] for trivial, [5] unsolved. I saw a professor who worked in differential topology for 40 years refer to this book -- and first year undergrads thumbing through the pages for exercises marked [1] and [2] to solve in spare time. This is a book for all levels of mathematicians: I am sure even the armchair amateur mathematicians can grasp some of the materials after a hard day's thought. I dont see this book as any less than a definitive text on enumerative combinatiorics.

8 of 8 people found the following review helpful:

5.0 out of 5 stars Very challenging, very deep, June 11, 2006

By 

Alexander C. Zorach "Alex Zorach" (Lancaster, PA) - See all my reviews
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This review is from: Enumerative Combinatorics, Volume 1 (Paperback)

This is an excellent book on combinatorics, but it is quite difficult to understand--written for experts, not novices. The author often chooses a more general framework in which to present things, and this can make the material quite difficult to follow. But the rewards for the diligent reader are great. Occasionally I question how Stanley chooses to present a certain topic, but usually if I look closely enough, I see that there are deep reasons for his choice of notation or presentation.

Some of the material in this book is easier than others; some of it depends on earlier chapters, but some stands on its own. People interested in partially ordered sets and lattices may want to jump ahead to that chapter--much of this chapter stands on its own, and it is an excellent exposition of that topic, and I think somewhat easier to understand than the rest of the book.

The most precious thing about this book is that the author manages to provide several comprehensive frameworks for solving large classes of enumeration problems. Combinatorics seems a hodge-podge subject to many mathematicians, but Stanley manages to see it as a unified subject with a number of general theories and common techniques. This book is truly the only text I have ever read that has this perspective on the subject.

I would recommend this book only to someone who has a strong background in mathematics and wants a challenging text that can take them to a deeper level of understanding. Students of combinatorics may want to take this book out of the library and read the introductory pages; there are some particularly useful comments right at the beginning. As a final note, the exercises in this book are also helpful and of diverse difficulty levels--and Stanley classifies the exercises by their difficulty level. People who find this book difficult to follow may want still benefit from some of the easier exercises. Students wanting an easier-to-follow text might want to check out Cameron's "Combinatorics", or Wilf's "Generatingfunctionology". As a final note I would like to remark that this book is very reasonably priced, especially when you consider the wealth of material it contains.

11 of 12 people found the following review helpful:

5.0 out of 5 stars A Masterpiece on Enumerative Combinatorics, January 27, 2005

By 

Aristarchus (San Diego, CA United States) - See all my reviews

This review is from: Enumerative Combinatorics, Volume 1 (Paperback)

I agree with the other reviewers. The book is a masterpiece on enumerative combinatorics. However, I am not so sure that it is a good book for a beginner. If you are a beginner, then you should read another book first, like John Riordan's book on "Combinatorial Analysis." Stanley's book is best suited for an advanced student who has a high level of mathematical mental maturity. The reason I say this is that in a few places Stanley's formalism, which is entirely appropriate for professional exposition, actually obscures the underlying simplicity of the mathematical ideas. We have all seen this in research papers, where a mathematician takes a trivial idea and "obsures" the underlying simplicity with too much formalism. However, for an advanced student, the book has a high density of important ideas and methods.