Dyson's Partition Ranks and their Multiplicative Extensions

TL;DR

This paper explores the properties of Dyson's rank function for partitions modulo 3, extends it multiplicatively, and identifies maximum values over all partitions of a given size.

Contribution

It introduces a multiplicative extension of Dyson's rank function and determines its maximum over all partitions of a fixed size, advancing understanding of partition rank distributions.

Findings

Convexity properties of Dyson's rank functions analyzed

Multiplicative extension of the rank function developed

Maximum rank values over partitions of size n identified

Abstract

We study the Dyson rank function $N(r,3;n)$, the number of partitions of $n$ with rank $\equiv r \pmod 3$. We investigate the convexity of these functions. We extend $N(r,3;n)$ multiplicatively to the set of partitions, and we determine the maximum value when taken over all partitions of size $n$.