Cyclic Derangements

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 qand (q, t)-analogs for cyclic derangements, generalizing results of Gessel, Brenti and Chow. 1 Derangements A derangement is a permutation that leaves no letter fixed. Algebraically, this is an element σ of the symmetric group Sn such that σ(i) 6= i for any i, or, equivalently, no cycle of σ has length 1. Geometrically, a derangement is an isometry in R of the regular (n − 1)-simplex that leaves no facet unmoved. Combinatorially, these are matrices with entries from {0, 1} such that each row and each column has exactly one nonzero entry and no diagonal entry is equal to 1. Let Dn denote the set of derangements in Sn, and let dn = |Dn|. The problem of counting derangements is the quintessential example of the principle of Inclusion-Exclusion [20]: