Cyclic derangements

TL;DR

This paper generalizes the concept of derangements to the wreath product of finite cyclic groups with symmetric groups, providing new enumeration formulas and q-analogues inspired by recent combinatorial research.

Contribution

It introduces a broad generalization of derangements in wreath products and develops q- and (q, t)-analogues, extending prior combinatorial results.

Findings

Derived enumeration formulas for cyclic derangements in wreath products

Established q-analogues of cyclic derangements

Extended combinatorial results to new algebraic structures

Abstract

A classic problem in enumerative combinatorics is to count the number of derangements, that is, permutations with no fixed point. Inspired by a recent generalization to facet derangements of the hypercube by Gordon and McMahon, we generalize this problem to enumerating derangements in the wreath product of any finite cyclic group with the symmetric group. We also give q- and (q, t)-analogs for cyclic derangements, generalizing results of Brenti and Gessel.