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

• Published in: arXiv.org
• Main Authors: Jin, Liuxin, Liu, Eric H, Xia, Ernest X W
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 18.03.2022
• Subjects: Congruences
• ISSN: 2331-8422

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 \(NT(m,j,n)\).