Identities from Partition of the Symmetric Group $${S\!}_n$$

Abstract

In this article, we define a combinatorial number, denoted by \(^nT_k\), and defined to be the number of permutations \(\sigma \in S_n\) which can be written as a product of k-number of transpositions but not m-number of transpositions for any integer m with \(1\leqslant m< k\). We establish a connection between this number and the Stirling number of the first kind and study some properties. Furthermore, we give a lexicographic order relation among the elements of the symmetric group on n-symbols and study, mainly, a concept, namely directional change of lexicographic permutations of \({S\!}_n\). The directional change of a permutation is a positive integer which indicates the hierarchy with its successive permutation in lexicographic ordering. We introduce a number, \(^n\mathbb {D}_{k}\) (for integers \(n\geqslant 2\)), which counts the number of lexicographic permutations of \({S\!}_n\) having the directional change value as k. We compute this number by means of recurrence relations, etc.