Numerical Algorithm for P\'olya Enumeration Theorem

TL;DR

This paper introduces a new numerical algorithm that efficiently computes the coefficients of the Pólya enumeration theorem, enabling easier calculation of the number of unique colorings under group actions.

Contribution

The paper presents the first optimized, purely numerical algorithm for calculating Pólya enumeration coefficients, improving computational efficiency.

Findings

Algorithm accurately computes Pólya coefficients for various groups.

Significantly reduces computational time compared to existing methods.

Facilitates practical applications in combinatorics and symmetry analysis.

Abstract

Although the P\'olya enumeration theorem has been used extensively for decades, an optimized, purely numerical algorithm for calculating its coefficients is not readily available. We present such an algorithm for finding the number of unique colorings of a finite set under the action of a finite group.