Arithmetic Properties of the Sequence of Derangements and its Generalizations

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.