Let d≥2 be an integer. In this paper we study arithmetic properties of the sequence (Hd(n))n∈N, where Hd(n) is the number of permutations in Sn 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 their properties. Among many results we obtain qualitative description of the p-adic valuation of the number Hd(n) extending in this way earlier results of Ochiai and Ishihara, Ochiai, Takegehara and Yoshida.