More Andrews–Beck type congruences by involutions

In 2021, Andrews studied the generating function for [Formula: see text], where [Formula: see text] denotes the total number of parts in the partitions of [Formula: see text] with rank congruent to [Formula: see text] modulo [Formula: see text], and then proved two congruences for [Formula: see text] conjectured by Beck. Later, Chern introduced [Formula: see text], which denotes the total number of ones in the partitions of [Formula: see text] with crank congruent to [Formula: see text] modulo [