What Is Enumerative Combinatorics?
1.1 How to Count
The basic problem of enumerative combinatorics is that of counting the number of elements of a finite set. Usually we are given an infinite collection of finite sets Si where i ranges over some index set I (such as the nonnegative integers ), and we wish to count the number f(i) of elements in each Si “simultaneously.” Immediate philosophical difficulties arise. What does it mean to “count” the number of elements of Si? There is no definitive answer to this question. Only through experience does one develop an idea of what is meant by a “determination” of a counting function f(i). The counting ...