A rank of partitions with overline designated summands

Abstract: 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 $PD_t(3n+2)$.