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Can anyone recommend a good introductory textbook on analytic combinatorics for self studying? It should be for a first year graduate student.

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Related: math.stackexchange.com/questions/15201/… –  lewellen Jun 18 '11 at 19:04
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9 Answers

One book I'd highly recommend is Peter J. Cameron's

Combinatorics: Topics, Techniques, Algorithms

The first link above is to site for the book, which includes multiple resources, including links, solutions to problems (good for self-study, etc.), additional exercises and projects. I used it in an early graduate "Special Topics" class on Combinatorics. (It's geared to be "basic" for grads, advanced for undergrads. Indeed, the text is designed in such a way as to provide ample material for a two-semester sequence, for self-study, and as a reference for future needs.) I have since self-studied the text on my own, to cover material which simply could not be covered in a single semester.

The text is loaded, and expansive in its coverage: dealing not only with combinatorial "content", but also combinatorial techniques, proofs, as well as algorithms for those with computational interest in combinatorics etc...and Cameron himself suggests which chapters to study for different aims.

You can peruse/preview the book at Amazon: Take a look inside, and also through the top link.

Table of Contents

1. What is combinatorics?
2. On numbers and counting
3. Subsets, partitions, permutations
4. Recurrence relations and generating functions
5. The principle of inclusion and exclusion
6. Latin squares and SDRs
7. Extremal set theory
8. Steiner triple theory
9. Finite geometry
10. Ramsey's theorem
11. Graphs
12. Posets, lattices and matroids
13. More on partitions and permutations
14. Automorphism groups and permutation groups
15. Enumeration under group action
16. Designs
17. Error-correcting codes
18. Graph colourings
19. The infinite
20. Where to from here?
Answers to selected exercises

Also, for a sample of Cameron's writing style, here's a pdf format of his Combinatorics Notes (also available at Cameron's home page, which is accessible from the book's site to which I've included a link at the top this post).

One other text that might suit your needs is Introduction to Combinatorial analysis by John Riordan, considered by many to be a "classic*.

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Richard Stanley has made his Enumerative Combinatorics, volume 1, available free for personal use (for awhile) on his website at http://math.mit.edu/~rstan/ec/ec1/. There's a new edition coming out and he's looking for possible corrections.

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That's a classic!! –  amWhy Jun 18 '11 at 22:59
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A Path to Combinatorics for Undergraduates: Counting Strategies [Paperback] Titu Andreescu (Author), Zuming Feng (Author)

Authors take a problem and start solving it. You start following solution and then all of a sudden they say "Not so fast my friend.. not so fast..." pointing out how that logic was faulty and then give the actual solution. This gives me kicks.

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Talk about fighting with a book! Multiple choice tests usually have several wrong answers that result from common mistakes on them but that takes it to a whole new level. ha! –  Tony Jun 19 '11 at 4:15
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Two staples are Concrete Mathematics and The Probabilistic Method.

But combinatorics is such a wide area, that you'll need to tell us what you want to focus on. For example, there is no "combinatorics on words" in any of the above.

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Check out Combinatorics through guided discovery by Kenneth T. Bogart. It is intended for self-study and introduces the concepts via a sequence of exercises which are integrated into each section. This way you are forced to digest each concept as it comes along, rather than browsing through the chapter and then tackling a list of problems. I highly recommend it.

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I guess Analytic Combinatorics by Flajolet and Sedgewick might be what you're looking for. It is freely available.

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I would not classify this book as introductory at all. –  Apprentice Queue Jun 18 '11 at 16:53
@apprentice: well, certainly not suitable for undergraduates, but not that hard either, I think… –  A. De Luca Jun 18 '11 at 19:33
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The books I always go back to are, in no particular order,

Ryser, Combinatorial Mathematics

Cohen, Basic Techniques of Combinatorial Theory

Tucker, Applied Combinatorics

Brualdi, Introductory Combinatorics

Comtet, Advanced Combinatorics.

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I highly recommend Fred Roberts' book Applied Combinatorics. (A new edition has appeared with Barry Tesman as co-author.)

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If you are looking for a tough but very good book I recommend the following. I am working through it right now: A Course in Combinatorics by Van Lint and Wilson:

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It gives a lot of proofs and it really has you thinking all the time.

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