2021, Andrews studied the generating function for NT(j, m, n), where NT(j, m, n) denotes the total number of parts in the partitions of n with rank congruent to j modulo m, and then proved two congruences for NT(j, m, n) conjectured by Beck. Later, Chern introduced M ω (j, m, n), which denotes the total number of ones in the partitions of n with crank congruent to j modulo m, and proved a list of more than 70 Andrews–Beck type congruences modulo 5, 7, 11 and 13 for NT(j, m, n) and M ω (j, m, n). In this paper, by means of some partition involutions, we find an efficient way to establish some new congruence relations between NT(j, m, n) and N(j, m, n) and the relations between M ω (j, m, n) and M(j, m, n), where N(j, m, n) (resp. M(j, m, n)) denotes the number of partitions of n with rank (resp. crank) congruent to j modulo m. As a consequence, we provide new proofs of the congruences in Chern’s list and find some new ones. Moreover, we extend this kind of congruence relations to overpartitions and k-marked Durfee symbols, and obtain more Andrews–Beck type congruences.
Gu et al. (Fri,) studied this question.