A NOTE ON STIRLING NUMBERS OF FIRST KIND AND CYCLE TYPES OF A PERMUTATION

Abstract

This paper proposes to establish a relationship between the Stirling numbers of the first kind and the cycle types of S_n, exhibiting the feasibility of a procedure to generate Stirling numbers of the first kind and proving some identities by the combination of these two concepts. This is possible due these numbers’ strong algebraic appeal, given that we can define them as the number of permutations of S_n that decompose into exactly k disjoint cycles. There is a bijective relationship between the cycle types of S_n and the partitions of a positive integer n, thus given a partition of n, we know how many permutations of S_n exist that are of a given cycle type. Given that all permutation of S_n can be decomposed into product of cycles, so we know how the number of permutations with a certain cycle type by looking at the integer partitions of n. Thus, Stirling numbers of the first kind can be easily determined. Throughout the article, we will explore some identities concerning Stirling numbers of the first kind and the binomial coefficient, as well as presenting the concepts of partitioning positive integers and cycle types of S_n.