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I have been studying number theory for a little while now, and I would like to learn about integer partitions and q series, but I have never studied anything in the field of combinatorics, so are there any prerequisites or things I should be familiar with before I try to study the latter?

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Could you make the title more specific? –  Rahul Jan 21 '13 at 4:27

7 Answers 7

George Andrews has contributed greatly to the study of integer partitions. (The link with his name will take you to his webpage listing publications, some of which are accessible as pdf documents.) Also see, e.g., his classic text The Theory of Partitions and the more recent Integer Partitions.

You can pretty much "jump right in" with the following, though their breadth of coverage may be more than you care to explore (in which case, they each have fine sections on the topics of interest to you, with ample references for more in depth study of each topic):

Two books I highly recommend are

Concrete Mathematics by Graham, Knuth, and Patashnik.

Combinatorics: Topics, Techniques, and Algorithms by Peter J. Cameron. See his associated site for the text.

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George Andrews and Kimmo Eriksson, Integer Partitions, is a very nice book about the topics you want to learn about. It says it requires nothing more of the reader "than some familiarity with polynomials and infinite series".

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I second this strongly, it's a lovely book. –  Chris Godsil Jan 18 '13 at 2:36
A great book I have used several times in upper division undergraduate seminar courses. –  Brian Hopkins Jan 23 '13 at 4:12

You should look into "Additive combinatorics" by Terence Tao. It establishes the connection between the two.

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For a more difficult introduction to some of the real magic of $q$-series, try "Number Theory in the Spirit of Ramanujan" by Bruce C. Berndt.

Even if you don't understand the proofs (my condition for much of it), the theorems are amazing.

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Fortunately a first look at combinatorics doesn't require very many prerequisites past high school mathematics. But generating functions are really cool, so if you know your power series from single-variable calculus that's great. Elementary linear algebra could also be useful.

If you'd like to go much deeper into combinatorics, like algebraic combinatorics or such, then you could go into abstract algebra and analysis as well. It's also nice to do graph theory alongside combinatorics (if you distinguish them at all).

I don't know much in the way of combinatorics resources, but if you end up liking generating functions then H. Wilf has made his book Generatingfunctionology available for free online. By all accounts it is one of the best written resources on the topic.

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Add "Analytic Combinatorics" by Flajolet and Segdewick (tough going) or "Introduction to the Analysis of Algorithms" by Segdewick and Flajolet (essentially the above, but easier to wrap your head around). The symbolic method they expose is elegant and simple to use. Both books are available for free at –  vonbrand Jan 21 '13 at 3:21

I recently came across the book How to count: An introduction to Combinatorics. It is an excellent book to learn combinatorics through self study from scratch. It has lots of solved problems and applications discussed in a very nice manner. Plus it has two chapters on partitions so that you can get an introduction to partition theory too. You can then move on to George Andrews.

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Other than Berndt's excellent book mentioned by marty cohen there is a book on $q$-series by Chan called An Invitation to Q-Series: From Jacobi's Triple Product Identity to Ramanujan's "Most Beautiful Identity". I haven't had the opportunity to study it yet, but it looks nice!

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