Enumerative Combinatorics, Volume 2 [Paperback]

Richard P. Stanley (Author), Sergey Fomin (Contributor)

Book Description

Publication Date: June 4, 2001 | ISBN-10: 0521789877 | ISBN-13: 978-0521789875 | Edition: 1

This second volume of a two-volume basic introduction to enumerative combinatorics covers the composition of generating functions, trees, algebraic generating functions, D-finite generating functions, noncommutative generating functions, and symmetric functions. The chapter on symmetric functions provides the only available treatment of this subject suitable for an introductory graduate course on combinatorics, and includes the important Robinson-Schensted-Knuth algorithm. Also covered are connections between symmetric functions and representation theory. An appendix by Sergey Fomin covers some deeper aspects of symmetric function theory, including jeu de taquin and the Littlewood-Richardson rule. As in Volume 1, the exercises play a vital role in developing the material. There are over 250 exercises, all with solutions or references to solutions, many of which concern previously unpublished results. 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"

"As a researcher, Stanley has few peers in combinatorics...the trove of exercises with solutions will form a vital resource; indeed, exercise 6.19 on the Catalan numbers, in 66 (!) parts, justifies the investment by itself. Both volumes highly recommended for all libraries."
Choice

"Volume 2 not only lives up to the high standards set by Volume 1, but surpasses them... Stanley's book is a valuable contribution to enumerative combinatorics. Beginners will find it an accessible introduction to the subject, and experts will still find much to learn from it."
Mathematical Reviews

Book Description

This is the second volume of a two-volume work on the subject of enumerative combinatorics, an area of mathematics with connections to many other topics within and outside of mathematics, such as computer science, spectroscopy, algebraic geometry, algebraic topology, and representation theory. Many topics covered (in particular, the theory of symmetric functions) are not available in any other textbook at this level, and the usefulness of the book is enhanced by over 250 exercises with solutions.Although primarily intended as a textbook for graduate students and a resource for professional mathematicians, some parts of the book will be accessible to mathematics undergraduates and even interested amateurs.

See all Editorial Reviews

Product Details

• Paperback: 600 pages
• Publisher: Cambridge University Press; 1 edition (June 4, 2001)
• Language: English
• ISBN-10: 0521789877
• ISBN-13: 978-0521789875
• Product Dimensions: 8.9 x 6 x 1.4 inches
• Shipping Weight: 1.8 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: #386,640 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.