1. What is Combinatorics?
Combinatorics is the slums of topology.
J. H. C. Whitehead (attr.)1
I have to admit that he was not bad at combinatorial analysis — a branch, however, that even then I considered to be dried up.
Stanislaw Lem, His Master’s Voice (1968)
Combinatorics is special. Most mathematical topics which can be covered in a lecture course build towards a single, well-defined goal, such as Cauchy’s Theorem or the Prime Number Theorem. Even if such a clear goal doesn’t exist, there is a sharp focus (finite groups, perhaps, or non-parametric statistics). By contrast, combinatorics appears to be a collection of unrelated puzzles chosen at random.
Two factors contribute to this. First, combinatorics is broad rather than deep. Its tentacles ...