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Euler's difference table associated to the sequence $\{n!\}$ leads naturallyto the counting formula for the derangements. In this paper we study Euler'sdifference table associated to the sequence $\{\ell^n n!\}$ and the generalizedderangement problem. For the coefficients appearing in the later table we willgive the combinatorial interpretations in terms of two kinds of $k$-successionsof the group $C_\ell\wr S_n$. In particular for $\ell=1$ we recover the knownresults for the symmetric groups while for $\ell=2$ we obtain the correspondingresults for the hyperoctahedral groups.

Derangements and Euler's difference table for $C_\ell\wr S_n$