New Identities Associated with Ranks and Cranks of Partitions Modulo 7

Abstract

Beck introduced two important partition statistics NT(r, m, n) and \(M_{\omega }(r,m,n)\) which count the total number of parts in the 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. Andrews confirmed two conjectures of Beck on congruences of NT(r, m, n). Inspired by Andrews’ work, Chern discovered a number of congruences modulo 5, 7, 11 and 13 of NT(r, m, n) and \(M_{\omega }(r,m,n) \). Recently, Mao, and Xia, Yan and Yao established several identities on NT(r, 7, n) and \(M_{\omega }(r,7,n)\) which yield some congruences modulo 7 due to Chern. Unfortunately, there are six congruences modulo 7 of Chern which are not implied by the identities given by Mao, and Xia, Yan and Yao. In this paper, we establish several new identities on NT(r, 7, n) and \(M_{\omega }(r,7,n)\). In particular, we prove six identities which are analogous to “Ramanujan’s most beautiful identity”and imply Chern’s six congruences.