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.