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

## TL;DR

This paper investigates the arithmetic properties of permutations composed of disjoint cycles of fixed length, including periodicity, divisibility, and p-adic valuations, extending previous research in the field.

## Contribution

It introduces new polynomial families related to these permutations and provides a comprehensive analysis of their arithmetic and divisibility properties, extending prior work.

## Key findings

- Characterization of periodicity modulo integers
- Analysis of p-adic valuations of permutation counts
- Identification of divisibility patterns and properties

## Abstract

Let $d\geq 2$ be an integer. In this paper we study arithmetic properties of the sequence $(H_d(n))_{n\in\N}$, 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.