The generalized symmetric group, or the group of colored permutations, is defined by the wreath product <?TeX $S(m,n)=\mathbb {Z}_m\wr S_n$?> Math 1 , where <?TeX $\mathbb {Z}_m$?> Math 2 denotes the cyclic group of order m and Sn denotes the symmetric group of degree n. We extend the notion of permutation classes, downward closed sets under containment, to colored permutations. The enumeration and growth rate of permutation classes have long been studied. For instance, if <?TeX $\mathcal {C}$?> Math 3 denotes a permutation class, then <?TeX $|\mathcal {C}\cap S_n|< F_n$?> Math 4 for some n, if and only if <?TeX $|\mathcal {C}\cap S_n|$?> Math 5 is eventually polynomial. This result is known as the Fibonacci dichotomy. We prove that the Fibonacci dichotomy also holds for colored permutation classes and extend the known algorithm that produces the polynomial in the case of permutations to colored permutations. We implement the algorithm and include applications to classes of colored permutations that require a fixed number of prefix reversals or block reversals to be sorted.

In this paper, we present an algorithm that enumerates a certain class of signed permutations, referred to as grid signed permutation classes. In the case of permutations, the corresponding grid classes are of interest because they are equivalent to the permutation classes that can be enumerated by polynomials. Furthermore, we apply our results to genome rearrangements and establish that the number of signed permutations with fixed prefix reversal and reversal distance is given by polynomials that can be computed by our algorithm.