A modular approach to Andrews-Beck partition statistics

Abstract

Andrews recently provided a q-series proof of congruences for $NT(m,k,n)$, the total number of parts in the partitions of n with rank congruent to m modulo k. Motivated by Andrews' works, Chern obtain congruences for $M_{\omega }(m,k,n)$ which denotes the total number of ones in the partition of n with crank congruent to m modulo k. In this paper, we focus on the modular approach to these new partition statistics. Applying the theory of mock modular forms, we establish equalities and identities for $NT(m,7,n)$ and $M_{\omega }(m,7,n)$.