A partition formula from idempotents

TL;DR

This paper derives a new partition formula using Burnside's Lemma applied to idempotent maps, involving elementary functions and a summation related to partition numbers, with discussions on potential elimination of the summation.

Contribution

It introduces a novel partition formula based on group action techniques and explores ways to simplify the summation involved.

Findings

Derived a partition formula involving elementary functions

Connected the formula to Burnside's Lemma and idempotent maps

Discussed potential methods to eliminate complex summations

Abstract

A formula which only involves a partition number and elementary functions is derived by applying Burnside's Lemma to the set of idempotent maps from a set to itself. One side involves a summation over a set closely related to the partition number, however. Some speculation is made as to how to eliminate this summation.