What Is Combinatorics? (2013)

There are a few quite large sub-fields, but there are classic textbooks covering most of these. Here are some:

For classical enumerative combinatorics, there is Richard Stanley's 'Enumerative Combinatorics'.

For graph theory, Diestel's book 'Graph Theory' is popular for introductory courses, and Bollobas' 'Modern Graph Theory' is more advanced and (as the title says) modern.

And there is Laszlo Lovasz's fantastic 'Combinatorial Problems and Exercises', which is a collection of problems and answers, and is a great way to get a feel for the subject.

Sagan's book on the symmetric group is a great introduction to algebraic combinatorics, tracing the combinatorial approach to understanding the representation theory of the symmetric group. For more depth in that direction, Fulton & Harris' 'Representation Theory' builds the connections with Lie groups.

Algebraic combinatorics dances in the intersection of a number of fields of mathematics, where some of the most fundamental and beautiful structures are to be found. I had a relatively large breadth of knowledge coming out of my phd as a result, which I'm pretty happy about.

http://www.amazon.com/Combinatorial-Theory-Marshall-Hall/dp/... ?