Arithmetic Properties of the Sequence of Derangements and its Generalizations

TL;DR

This paper explores the arithmetic properties of derangement sequences and their generalizations, including periodicity, valuations, and prime divisors, extending classical combinatorial results to broader classes of sequences.

Contribution

It introduces two new classes of sequences generalizing derangements and analyzes their arithmetic properties, such as periodicity, valuations, and recurrence relations.

Findings

Derangement sequences exhibit specific periodicity modulo d.

p-adic valuations of these sequences are characterized.

Prime divisors of the sequences are identified and analyzed.

Abstract

The sequence of derangements is given by the formula $D_0 = 1, D_n = nD_{n-1} + (-1)^n, n>0$. It is a classical object appearing in combinatorics and number theory. In this paper we consider two classes of sequences: first class is given by the formulae $a_0 = h_1(0), a_n = f(n)a_{n-1} + h_1(n)h_2(n)^n, n>0$, where $f,h_1,h_2 \in\mathbb{Z}[X]$, and the second one is defined by $a_n = \sum_{j=0}^n \frac{n!}{j!} h(n)^j, n\in\mathbb{N}$, where $h\in\mathbb{Z}[X]$. Both classes are a generalization of the sequence of derangements. We study such arithmetic properties of these sequences as: periodicity modulo $d$, where $d\in\mathbb{N}_+$, $p$-adic valuations, asymptotics, boundedness, periodicity, recurrence relations and prime divisors. Particularly we focus on the properties of the sequence of derangements and use them to establish arithmetic properties of the sequences of even and odd derangements.