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I'm currently attending a somewhat disorganized seminar on combinatorics that follows no textbook. So far we have covered the orbit-stabilizer theorem, some recursion, and we're heading into the Möbius inversion formula.

Can anyone suggest a text that approaches combinatorics at this level for a 2nd-3rd year undergrad who already knows some algebra and the more basic combinatorics like combinations, permutations, stars-and-bars, generating functions? Most introductory combinatorics books I've found are more suited to a discrete math class and cover stuff which I already know. I'm looking for something to supplement this lecture. Thank you.

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You would probably be interested in… – BBischof Feb 12 '11 at 5:59

I can't say I can predict what direction your seminar will head in, but an amazing one-stop-shop for combinatorics is Richard Stanley's Enumerative Combinatorics (both volumes). You will learn a tremendous amount if you get through both of these (and the revised version of volume 1 is currently available online!). In particular Volume I has a thorough discussion of Möbius inversion.

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I think Stanley's Enumerative Combinatorics is not an "intermediate" level book, it's the canonical advanced book on combinatorics.

I'd recommend Aigner's "A Course on Enumeration" (from Springer's GTM series) for a lighter level (but certainly beyond apples and oranges) go A Walk Through Combinatorics (Miklos Bona) or Combinatorics: Topics, Techcniques and Algorithms (Cameron)

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I've also had good luck with van Lint's and Wilson's Course in Combinatorics. I am finding that reading a section from van Lint, then the section from Aigner, then the section from Stanley is pretty steady going. I'll check out Bona and Cameron. – Jack Schmidt Apr 5 '11 at 19:33
Bona also has a book at a higher level than "A Walk Through Combinatorics", entitled "Introduction to Enumerative Combinatorics". I don't know this book but if it's in your library it's probably worth looking at. – Michael Lugo Apr 5 '11 at 21:01
For the OP, Cameron's book looks the closest. It covers most of the general topics (graphs, designs, enumeration, arrangements) at an intermediate level. It is very accessible/readable, but not elementary. The technique of using more than one textbook is an excellent one though. – Mitch Apr 5 '11 at 21:10
Yes, Introduction to Enumerative Combinatorics is great too, but as you point, a bit harder. I find for example, generating funcions and matchings are better covered here that on Walkthrough, but it quickly goes into specialized topics you might not be as interested unless you're particularly into combinatorics – Drini Apr 7 '11 at 5:27

I did a reading course in Combinatorics while a PhD student, and we used van Lint and Wilson which I thought was very hard, but very good. You don't need any more background than you already have, and you will learn a ton from this text.

It's definitely not as well-known as the other suggested texts, but it many ways it's superior. (Although Concrete Mathematics is a better book overall for its sheer beauty.)

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I second van Lint and Wilson. It is structured as a menu of stand alone topics, but hits the full range of them (but at a little past intermediate level). – Mitch Apr 5 '11 at 21:06

If you read French, you must read Analyse Combinatoire of L. Comtet at PUF edition. It's two short books full of fascinating materials.

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In English it's Advanced Combinatorics and is definitely worth a look. – Mike Spivey Feb 12 '11 at 2:51

How about Concrete Mathematics?

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He said he knows about combinations to generating functions. So I skip it. But I'm not sure how one can master all this book in undergrad (or after also ...) – Sam Feb 12 '11 at 0:43
I love GKP's 'Concrete Mathematics' but it covers a very particular subarea of combinatorics, and frankly is more like applied algebraic manipulation. That is, it is a very narrow view of combinatorics, hardly a mention of graphs, designs, or partial orders. – Mitch Apr 5 '11 at 21:05

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