Andrews-Beck type congruences modulo arbitrary powers of 5 for 2-colored partitions

Abstract

Recently, Andrews proved two conjectures on a partition statistic introduced by Beck. 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 Toh introduced a partition statistic of k-colored partitions [Formula: see text] which counts the total number of parts of the first component in each k-colored partition [Formula: see text] of n with [Formula: see text] congruent to r modulo m and proved many congruences for [Formula: see text]. Very recently, Du and Tang proved a number of Andrews–Beck type congruences for [Formula: see text] and confirmed all conjectures posed by Lin, Peng and Toh. Motivated by their work, we establish the generating functions of [Formula: see text] and prove several families of congruences modulo arbitrary powers of 5 for [Formula: see text]. In particular, we generalize a congruence modulo 5 for [Formula: see text] due to Lin, Peng and Toh.