Recently, Beck introduced two partition statistics NT(r, m, n) and \(M_{\omega }(r,m,n)\) , which count the total number of parts in partitions of n with rank congruent to r modulo m and the total number of ones in the partitions of n with crank congruent to r modulo m, respectively. He also posed two conjectures on NT(r, m, n) which were confirmed by Andrews. Very recently, Chern discovered a number of Andrews-Beck type congruences on NT(r, m, n) and \(M_{\omega }(r,m,n)\) and some of them were conjectured by Chan, Mao and Osburn. Inspired by Chern’s work, we prove many new Andrews-Beck type congruences modulo 7, 11 and 13 on linear combinations of NT(r, m, n) and \(M_{\omega }(r,m,n)\) in this paper.
