  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 $PD_{t}(n)$ which counts the number of tagged
parts over all the partitions of $n$ with designated summands. He proved that
$PD_{t}(3n+2)$ is divisible by $3$. In this paper, we first introduce a
structure named partitions with overline designated summands, which is counted
by $PD_t(n)$. We then define a generalized rank of partitions with overline
designated summands and give a combinatorial interpretation of the congruence
for $PD_t(3n+2)$.
