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 PDt(n) which counts the number of tagged parts over all the partitions of n with designated summands. He proved that PDt(3n + 2) is divisible by 3. In this paper, we first introduce a structure named partitions with overline designated summands, which is counted by PDt(n). We then define a generalized rank of partitions with overline designated summands and give a combinatorial interpretation of the congruence for PDt(3n +2). (C) 2021 Elsevier B.V. All rights reserved.