Customer Reviews

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

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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.

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.

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.

I am what one of the reviewers called an arm-chair mathematician..

While I do not believe, that this book is suitable as a first introduction to combinatorics, it is a great book for anyone, who is interested in the subject and has had some prior exposure.
The formal mathematical prerequisites are quite minimal. The proofs are such, that after some thinking one can understand them - and they are always rigorous.
If one some occasions, the author would have given a short hint, instead of simply saying "it is easily seen" this would have made the book even more readable (but even then, after enough thinking one does see, albeit maybe not easily...)

All in all, very recommendable
(I am referring to volume 1 only, I did not read volume 2)

I have now ( March 2011) also read volume II
Once again, the formal prerequisites are not very high, but the proofs are definitely harder to understand than in volume I! While chapters 5&6 are still not too difficult, chapter 7 ( about symmetric functions) certainly is hard. A few additional lines of explanation would have made the proofs easier to understand - at least for an "ar chair mathematician"!
The book contains mainy excercises with solutions(!!!), which I did not attempt, so I cannot comment on them.

The book is virtually free of any typos, I counted maybe 10 harmless ones.

I would still recommend volume II to anybody seriously interested in combinatorics, but it is definitely harder than volume I

Overall, my rating remains at "full score"!

This book is a must for anyone who likes how to count. In addition to the superb exposition of deep and important mathematics, it contains so many intriguing problems, some of them even puzzle-like. Read this book cover-to-cover or open it at a random page. Either way you would love it!

There was an earier review that claimed this book is for "people who like to count." That's a little silly. This book is a rigorous math text. And it's glorious. It's probably my favorite text. But it's not light reading at all.

I spent a semester actively reading and working on this book with my advisor. I read this book and worked on research, 50/50 split on my time. I got through 2.5 of the 4 chapters, and I'm damn proud of myself. It's a great book, but if you didn't know that 'enumerative' was for "people who like to count", you probably want a different text.