Some identities on Lin-Peng-Toh's partition statistic of $k$-colored partitions

Abstract

Recently, Andrews proved two conjectures on a partition statistic introduced by Beck. Very recently, Chern established some results on weighted rank and crank moments and proved many Andrews-Beck type congruences. Motivated by Andrews and Chern's work, Lin, Peng and To introduced a partition statistic of $k$-colored partitions $NB_k(r,m,n)$ which counts the total number of parts of $\pi^{(1)}$ in each $k$-colored partition $\pi$ of $n$ with ${\rm crank}_k(\pi)$ congruent to $r$ modulo $m$ and proved a number of congruences for $NB_k(r,m,n)$. In this paper, we prove some identities on $NB_k(r,m,n)$ which are analogous to Ramanujan's ``most beautiful identity". Moreover, those identities imply some congruences proved by Lin, Peng and Toh.