Some identities for Lin–Peng–Toh’s k-colored partition statistic

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