Some inequalities and equalities on Lin–Peng–Toh’s partition statistic for k-colored partitions

A k-colored partition \(\pi \) of a positive integer n is a k-tuple of partitions \(\pi =(\pi ^{(1)},\ldots , \pi ^{(k)})\) such that \(|\pi ^{(1)}| +\cdots +|\pi ^{(k)}|=n\). Recently, Fu and Tang defined a generalized crank for k-colored partitions by \( \textrm{crank}_k(\pi ) =\#(\pi ^{(1)})-\#(\pi ^{(2)}) \), where \(\#(\pi ^{(i)})\) denotes the number of parts in \(\pi ^{(i)}\). They also proved some inequalities and equalities for \(M_k(m,j,n)\) which counts the number of k-colored partitions of n with generalized crank congruent to m modulo j. Very recently, Lin, Peng and Toh established some new Andrews–Beck type congruences on \(NB_k(m,j,n)\) which denotes the total number of parts of \(\pi ^{(1)}\) in each k-colored partition \(\pi \) of n with \( \textrm{crank}_k(\pi )\) congruent to m modulo j. In this paper, motivated by the work of Fu–Tang and Lin–Peng–Toh, we establish the generating functions for \(NB_k(m,j,n)\) when \(j=2,3,4\) and deduce some new inequalities and equalities for \(NB_k(m,j,n)\).