Colorful Rings of Partition

TL;DR

This paper explores visualizations of the partition function p(n), connecting various mathematical fields, and introduces novel ring structures of partitions to deepen understanding of number theory concepts.

Contribution

It introduces finite and infinite rings of partitions and discusses their role in visualizing and understanding the partition function and related number theory results.

Findings

Visualization of p(n) using rings enhances understanding.

Connections between different mathematical fields are illustrated.

New ring structures provide insights into partition identities.

Abstract

We visualize the identity p(n) = sum s(k) p(n-k)/n for the integer partition function p(n) involving the divisor function s, add comments on the history of visualizations of numbers, illustrate how different mathematical fields play together when proving lim p(n)^(1/n)=1 and introduce finite or infinite rings of partitions.