# Partition function of the cyclic group

## TL;DR

This paper explores the enumeration of partitions of elements in cyclic groups into distinct parts, establishing a link to bi-color necklaces with specific periodicity, extending known results to even moduli.

## Contribution

It generalizes the connection between group partitions and bi-color necklaces to include even moduli, expanding previous odd-modulus results.

## Key findings

- Established equivalence between $Q_{m,t}(n)$ and bi-color necklaces with periodicity constraints for even $m$
- Extended known results from odd to even cyclic group moduli
- Provided combinatorial interpretations for group partitions

## Abstract

This paper addresses the problem of finding $Q_{m,t}\left(n\right)$, the number of possible ways to partition any member $n$ of the cyclic group $\mathbb{Z}/m\mathbb{Z}$ into $t$ distinct parts. When $m$ is odd, it was previously known that the number of partitions of the identity element $0\bmod m$ with distinct parts is equal to the number of possible bi-color necklaces with $m$ beads. This paper will expand upon this result by showing the equivalence between $Q_{m,t}\left(n\right)$ and the number of bi-color necklaces meeting certain periodicity requirements, even when $m$ is even.