Integer partitions with large Dyson rank

TL;DR

This paper investigates the distribution of Dyson ranks in integer partitions, providing formulas and identities for partitions with large ranks and specific residue class counts, enhancing understanding of partition rank properties.

Contribution

It introduces formulas and identities for counting partitions with large Dyson ranks and fixed residue classes, using Fine-Dyson symmetry, which is a novel approach.

Findings

Formulas for partitions with rank > n/2

Identities for partitions with large rank in fixed residue classes

Enhanced understanding of Dyson rank distribution

Abstract

The Dyson rank of an integer partition is the difference between its largest part and the number of parts it contains. Using Fine-Dyson symmetry, we give formulas for the number of partitions of n with rank larger than n/2, and we prove identities for counts of partitions with large rank in fixed residue classes.