ABSTRACT
Today, combinatorics is an important branch of mathematics and has many applications in computer science, physics, chemistry, and biology. Usually, one is interested in counting objects of a set that depend on a parameter or several parameters, for example, the number of partitions of the set [n] (here the parameter is n), or the number of set partitions of the set [n] with exactly k blocks (here the parameters are n and k). It is not hard to enumerate the number of such partitions for small values of n and k by exhibiting all possibilities, but as n (k) increases, the number of such partitions grows very fast, and so we need to be smarter about enumeration. In this chapter, we will give an overview of some basic techniques and ideas of combinatorics. The main goal of this chapter is to provide and to overview 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 elementary examples. The proofs of these facts and ideas can be found in any book on combinatorics; for example, see [230, 506, 935, 1036].
