TITLE: Generating functions for combinatorics
QUESTION [21 upvotes]: I have no formal education in generating functions, but, based on another question, I have seen that they can be powerful for combinatorics.
Are there a few general principles for using generating functions to construct counting arguments (e.g. counting the number of subsets or number of total orderings) that can be remixed to count different quantities?
If the answer is not amenable to this forum, where can I learn more? 
I've been trying to read Concrete Mathematics (Graham, Knuth, Patashnik), but the topic spans several chapters and over 100 pages. At the moment, I just have patience to learn the gist of the combinatorics applications of generating functions. Maybe later I'll go back and get an extensive education from GKP.

REPLY [9 votes]: Flajolet and Sedgewick's Analytic Combinatorics is dense, but equips you with amazing tools to construct, manipulate, and extract information from generating functions. I also found the relevant chapters of Stanley's Enumerative Combinatorics (both volumes) extremely helpful.
Generating functions happen to be a favorite topic of mine, so I've written several posts on the subject on my blog. You might want to start at this post, which builds to an explanation of the Polya enumeration theorem, and a few years ago I wrote up these notes which might be useful. 
There are in fact a few general principles, but I have enormous difficulty concisely describing them.