Restricted k-color partitions, II

We consider (k, j)-colored partitions, partitions in which k colors exist but at most j colors may be chosen per size of part. In particular these generalize overpartitions. Advancing previous work, we find new congruences, including in previously unexplored cases where k and j are not coprime, as well as some noncongruences. As a useful aside, we give the apparently new generating function for the number of partitions in the N x M box with a given number of part sizes, and extend to multiple colors a conjecture of Dousse and Kim on unimodality in overpartitions.