  In this paper, we define a set which has a finite group action and is
generated by a finite color set, a set which has a finite group action, and a
subset of the set of non negative integers. we state its properties to apply
one of solution of the following two problems, respectively. First, we
calculate the generating function of the character of symmetric powers of
permutation representation associated with a set which has a finite group
action. Second, we calculate the number of primitive colorings on some objects
of polyhedrons. It is a generalization of the calculation of the number of
primitive necklaces by N.Metropolis and G-C.Rota.

Combinatorics;Representation Theory