# Symmetric powers of permutation representations of finite groups and primitive colorings on polyhedrons

## TL;DR

This paper develops methods to compute generating functions for symmetric powers of permutation representations and counts primitive colorings on polyhedrons, generalizing previous work on primitive necklaces.

## Contribution

It introduces a framework for analyzing symmetric powers of permutation representations and primitive colorings, extending classical results to more complex structures.

## Key findings

- Derived formulas for generating functions of symmetric powers.
- Calculated the number of primitive colorings on polyhedrons.
- Generalized the enumeration of primitive necklaces.

## Abstract

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.