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

Abstract

Recently, George Beck introduced two partition statistics N T (m, j, n) and M ω (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 N T (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 N T (m, j, n) and M ω (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 ω (m, j, n) and N T (m, j, n).