We’ve got everything we need to prove the Polya enumeration theorem. To **state** the theorem, however, requires the language of generating functions, so I thought I’d take the time to establish some of the important ideas. It isn’t possible to do justice to the subject in one post, so I’ll start with some references.

Many people recommend Wilf’s generatingfunctionology, but the terminology is non-standard and somewhat problematic. Nevertheless, it has valuable insight and examples.

I cannot recommend Flajolet and Sedgewick’s Analytic Combinatorics highly enough. It is readable, includes a wide variety of examples as well as very general techniques, and places a great deal of emphasis on asymptotics, computation, and practical applications.

If you can do the usual computations but want to learn some theory, Bergeron, Labelle, and Laroux’s Combinatorial Species and Tree-like Structures is a fascinating introduction to the theory of species that requires fairly little background, although a fair amount of patience. It also contains my favorite proof of Cayley’s formula.

Doubilet, Rota, and Stanley’s On the idea of generating function is part of a fascinating program for understanding generating functions with posets as the fundamental concept. I may have more to say about this perspective once I learn more about it.

While it is by no means comprehensive, this post over at Topological Musings is a good introduction to the basic ideas of species theory.

And a shameless plug: the article I wrote for the Worldwide Online Olympiad Training program about generating functions is available here. I tried to include a wide variety of examples and exercises taken from the AMC exams while focusing on techniques appropriate for high-school problem solvers. There are at least a few minor errors, for which I apologize. You might also be interested in this previous post of mine.

In any case, this post will attempt to be a relatively self-contained discussion of the concepts necessary for understanding the PET.

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