Separable integer partition classes and partitions with congruence conditions

In this article, we first investigate the partitions whose parts are congruent to a or b modulo k with the aid of separable integer partition classes with modulus k introduced by Andrews. Then, we introduce the $(k,r)$-overpartitions in which only parts equivalent to r modulo k may be overlined and we will show that the number of $(k,k)$-overpartitions of n equals the number of partitions of n such that the k-th occurrence of a part may be overlined. Finally, we extend separable integer partition classes with modulus k to overpartitions and then give the generating function for $(k,r)$-modulo overpartitions, which are the $(k,r)$-overpartitions satisfying certain congruence conditions.