Twelvefold Way - Generalizations

Generalizations

We can generalize further by allowing other groups of permutations to act on N and X. If G is a group of permutations of N, and H is a group of permutations of X, then we count equivalence classes of functions . Two functions and are considered equivalent if, and only if, there exist so that . This extension leads to notions such as cyclic and dihedral permutations, as well as cyclic and dihedral partitions of numbers and sets.