A proof of some conjectures of Mao on partition rank inequalities

TL;DR

This paper proves several of Mao's conjectured inequalities related to partition ranks modulo 6 and 10, building on previous work and using elementary methods.

Contribution

It provides the first proofs of some of Mao's conjectured inequalities for partition ranks modulo 6 and 10, expanding understanding of partition rank inequalities.

Findings

Proved inequalities for partition ranks modulo 10.

Established results for the $M_2$ rank without repeated odd parts.

Used elementary methods for proofs.

Abstract

Based on work of Atkin and Swinnerton-Dyer on partition rank difference functions, and more recent work of Lovejoy and Osburn, Mao has proved several inequalities between partition ranks modulo $10$, and additional results modulo $6$ and $10$ for the $M_2$ rank of partitions without repeated odd parts. Mao conjectured some additional inequalities. We prove some of Mao's rank inequality conjectures for both the rank and the $M_2$ rank modulo $10$ using elementary methods.