A proof of some conjectures of Mao on partition rank inequalities

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.