What's a good advanced textbook/resource for studying the complexity of counting and combinatorics? I'm taking a class in enumerative combinatorics. The professor focusses on the complexity of solving combinatorics problems like partitions etc. I'm using Enumerative Combinatorics but it does not talk about complexity. What's a good book/website/resource that touches on enumerative (or even extremal) combinatorics and complexity of counting?
If I understood correctly, you're looking for a book (chapter) on counting complexity. In this case, I'd recommend the book of Sanjeev Arora and Boaz Barak, which has a chapter on counting complexity. A draft of it can be found here.

Concerning Extremal Combinatorics, Wiki suggests reading Extremal and Probabilistic Combinatorics by Noga Alon and Michael Krivelevich which contains pointers to some resources (mainly graph-related).