The number of tagged parts over the partitions with designated summands

We are concerned with two types of partitions considered by Andrews, Lewis and Lovejoy. One is the partitions with designated summands where exactly one is tagged among parts with equal size. The other is the partitions with designated summands where all parts are odd. In this paper, we study two partition functions PDt(n) and PDOt(n), which count the number of tagged parts over the above two types of partitions respectively. We first give the generating functions of PDt(n) and PDOt(n). Then we establish many congruences modulo small powers of 3 for them. Finally, we pose some problems for future work.