On the Number of Partitions of Sets and Natural Numbers

Partitions of sets and natural numbers have been a very attractive subject during the recent decades. Partitions play important roles in such diverse areas of mathematics as combinatorics, Lie theory, representation theory, mathematical physics, and the theory of special functions. Because of some applications of this subject, mathematicians have given some formulas in this regard. Up to now, the number of partitioning of sets and natural numbers was considered by some authors; for more details about partitions of sets and other related subjects, see for example [8], [10], [15], [17], [18], [19], [20], [21], and [22], and for partitions of natural numbers and other related concepts, see for example [1], [2], [3], [4], [5], [9], [12], [13], and [14]. In this article, using some elementary tools of combinatorial analysissee [6], [7], [11], [16], and [23]we give some alternative formulas for theses problems, considering also some special cases.