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.