A rank of partitions with overline designated summands

Andrews, Lewis and Lovejoy introduced the partition function $PD(n)$ as the number of partitions of $n$ with designated summands. In a recent work, Lin studied a partition function $P{D}_t(n)$ which counts the number of tagged parts over all the partitions of $n$ with designated summands. He proved that $P{D}_t(3n+2)$ is divisible by $3$. In this paper, we first introduce a structure named partitions with overline designated summands, which is counted by $P{D}_t(n)$. We then define a generalized rank of partitions with overline designated summands and give a combinatorial interpretation of the congruence for $P{D}_t(3n+2)$.