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 remaining 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).
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