Combinatorial Species

I've just discovered the most amazing piece of math- Combinatorial Species. I was aware of the "polynomial" types of Haskell and the notation, which itself is amazing, as well as the idea that we can take the derivative of a type to get the same type of thing, but with a "hole" in it.

But this is just one small part of the much larger theory being used here. The basic idea (as I understand it) is that in the field of combinatorics it has long been known that generating functions can be used to solve combinatorial problems. It is possible to perform operations on the simple, well known, series and find solutions to complex problems. There are quite a few operations, including a sum, product, composition, functorial composition, cartisian product, and differentiation. It turns out on the other hand, that we can apply operators to datatypes, as written in a particular notation, and get the structures involved in a problem, while simultaneously performing the analogous operations on the datatypes associated generating series, and both will tell us about the same problem.

I'm being very vague here about what exactly is going on with this theory, but I hope to post more about it, and hopefully figure more out myself.