Derangements and Euler's difference table for \(C_{l} \wr S_{n}\) (Q1010785)

Summary: Euler's difference table associated to the sequence \(\{n!\}\) leads naturally to the counting formula for the derangements. In this paper we study Euler's difference table associated to the sequence \(\{\ell^n n!\}\) and the generalized derangement problem. For the coefficients appearing in the later table we will give the combinatorial interpretations in terms of two kinds of \(k\)-successions of the group \(C_\ell\wr S_n\). In particular for \(\ell=1\) we recover the known results for the symmetric groups while for \(\ell=2\) we obtain the corresponding results for the hyperoctahedral groups.