On some properties of the number of permutations being products of pairwise disjoint $d$-cycles

Abstract

Let $d\ge 2$ be an integer. In this paper we study arithmetic properties of the sequence $(H_d(n){)}_{n\in \mathbb{N}}$, where $H_d(n)$ is the number of permutations in $S_n$ being products of pairwise disjoint cycles of a fixed length $d$. In particular we deal with periodicity modulo a given positive integer, behaviour of the $p$-adic valuations and various divisibility properties. Moreover, we introduce some related families of polynomials and study they properties. Among many results we obtain qualitative description of the $p$-adic valuation of the number $H_d(n)$ extending in this way earlier results of Ochiai and Ishihara, Ochiai, Takegehara and Yoshida.

35 pages