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

Yueya Hu, Eric H. Liu, Olivia X. M. Yao

• A k-colored partition π of a positive integer n is a k-tuple of partitions π = (π(1) ,...,π(k) ) such that |π(1) |+···+|π(k) | = n. Recently, Fu and Tang defined a generalized crank for k-colored partitions by crankk (π ) = #(π(1) ) − #(π(2) ), where #(π(i) ) denotes the number of parts in π(i) . They also proved some inequalities and equalities for Mk (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 N Bk (m, j, n) which denotes the total number of parts of π(1) in each k-colored partition π of n with crankk (π ) 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 N Bk (m, j, n) when j = 2, 3, 4 and deduce some new inequalities and equalities for N Bk (m, j, n).