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

TL;DR

This paper extends Euler's difference table to the sequence involving wreath products of cyclic groups and symmetric groups, providing combinatorial interpretations for generalized derangements and unifying known results for symmetric and hyperoctahedral groups.

Contribution

It introduces a new Euler difference table for the sequence \\{ll^n n!\\} and offers combinatorial interpretations for its coefficients in terms of k-successions in wreath product groups.

Findings

Derived formulas for generalized derangements in wreath product groups.

Connected results to classical derangements for symmetric and hyperoctahedral groups.

Provided combinatorial interpretations for new coefficients in the difference table.

Abstract

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.