**1**

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 *S*_{i} 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 *S*_{i} “simultaneously.” Immediate philosophical difficulties arise. What does it mean to “count” the number of elements of *S*_{i}? 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 ...