What is the difference between analytic combinatorics and the theory of combinatorial species?

Question

Yesterday I asked the question Why should a combinatorialist know category theory?, where Chris Taylor suggested me to have a look at combinatorial species. I had heard the term before but I haven't read more about it that than its entry on Wikipedia. To me it looks essentially like analytic combinatorics dressed in the language of categories. Is this correct? Can we do things with species that are impossible with analytic combinatorics (or vice versa)?

Answer 1

Analytic combinatorics tells you how to extract useful information from a generating function. The theory of combinatorial species is, among other things, an elegant way of writing down generating functions from a description of a combinatorial problem. So the two are complementary.

I recommend Bergeron, Labelle, and Leroux's Combinatorial Species and Tree-like Structures to really get a feel for the latter.

Answer 2

Analytic combinatorics uses generating functions that are power series of functions analytic in some region. The location of singularities and the residues or asymptotic behavior near the singularities are used to treat the asymptotics of the coefficients. These are non-formal calculations that cannot be done in any direct way with formal power series.

Species is an enriched version of the completely formal use of exponential generating functions. It is not necessary for the series to converge anywhere. Very roughly, it is finite counting, but keeping track of automorphisms or other "structure".

Answer 3

At the time of writing that Wikipedia article on Analytic combinatorics does not at all get into the meat of analytic combinatorics which is the complex analysis part. The article only covers the symbolic part which is the first half of the book on Analytic Combinatorics: http://algo.inria.fr/flajolet/Publications/book.pdf

The second half of the book is the analysis part about asymptotics mentioned in zyx's answer.