Proofs of some conjectures of Chan-Mao-Osburn on Beck's partition statistics

TL;DR

This paper confirms three conjectures by Chan-Mao-Osburn and two by Mao related to partition statistics, expanding understanding of congruences involving Beck's partition statistics and introducing new conjectures.

Contribution

The paper proves several conjectures on partition statistics $NT(m,j,n)$ and $M_{ ext{omega}}(m,j,n)$, advancing the theory of partition congruences and statistics.

Findings

Confirmed three conjectures of Chan-Mao-Osburn

Confirmed two conjectures of Mao

Proposed two new conjectures on partition statistics

Abstract

Recently, George Beck introduced two partition statistics $NT(m,j,n)$ and $M_{\omega}(m,j,n)$, which denote the total number of parts in the partition of $n$ with rank congruent to $m$ modulo $j$ and the total number of ones in the partition of $n$ with crank congruent to $m$ modulo $j$, respectively. Andrews proved a congruence on $NT(m,5,n)$ which was conjectured by Beck. Very recently, Chan, Mao and Osburn established a number of Andrews-Beck type congruences and posed several conjectures involving $NT(m,j,n)$ and $M_{\omega}(m,j,n)$. Some of those conjectures were proved by Chern and Mao. In this paper, we confirm the remainder three conjectures of Chan-Mao-Osburn and two conjectures due to Mao. We also present two new conjectures on $M_{\omega}(m,j,n)$ and $NT(m,j,n)$.