The classical derangement numbers count fixed point-free permutations. In this paper we study the enumeration problem of generalized derangements, when some of the elements are restricted to be in distinct cycles in the cycle decomposition. We find exact formula, combinatorial relations for these numbers as well as analytic and asymptotic description. Moreover, we study deeper number theoretical properties, like modularity, p-adic valuations, and diophantine problems.

The sequence of derangements is given by the formula D 0 = 1, Dn = 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, and the second one is defined by. Both classes are a generalization of the sequence of derangements. We study such arithmetic properties of these sequences as: periodicity modulo d, where d ∈ 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.

A prime number p is called a Schenker prime if there exists such n ∈ N + that p ∤ n and p | an, where an = n j=0 n! j! n j is so-called Schenker sum. T. Amdeberhan, D. Callan and V. Moll formulated two conjectures concerning p-adic valuations of an in case when p is a Schenker prime. In particular, they asked whether for each k ∈ N + there exists the unique positive integer n k < p k such that vp(a m·5 k +n k ) ≥ k for each nonnegative integer m. We prove that for every k ∈ N + the inequality v 5 (an) ≥ k has exactly one solution modulo 5 k . This confirms the first conjecture stated by the mentioned authors. Moreover, we show that if 37 ∤ n then v 37 (an) ≤ 1, what means that the second conjecture stated by the mentioned authors is not true.