ABSTRACT
Counting elements of a set is the heart of mathematics and embodies the magic of the combinatorics. Usually, we are interested in counting objects that depend on a parameter (or two, or three ...), for example finding the number of set partitions of [n] (here, the parameter is n) or finding the number of set partitions of [n] with exactly k blocks (here, the parameters are n and k). By exhibiting all possibilities of set partitions for small values of n, we find the number of these objects, but as n increases, we need to be smarter about counting. The main goal of this chapter is to provide and to overview of some basic facts and ideas of counting without proofs, where we will illustrate several techniques such as the use of recurrence relations, generating functions, and combinatorial bijections with very basic examples. The proofs of these basic facts and ideas can be founded in any basic book on combinatorics; for example, see the second chapter in [138] to find such proofs and other examples related to words and compositions.
