A new rank of partitions with overline designated summands

Robert. X. J. Hao, Erin Y. Y. Shen, Wenston J. T. Zang

Published 2020-04-07Version 1

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 the structure of partitions with overline designated summands, which is counted by $PD_t(n)$. We then define a generalized rank named $pdt$-rank on partitions with overline designated summands and provide a combinatorial interpretation for the congruence $PD_t(3n+2)\equiv 0\pmod{3}$. Let $N_{dt}(m,n)$ denote the number of partitions of $n$ with overline designated summands with $pdt$-rank $m$. We prove that $N_{dt}(m,n)\leq N_{dt}(m,n+1)$ with integer $m\not=0$, $n\geq 1$ as well as $N_{dt}(m,n)\leq N_{dt}(m+1,n)$ for $m\geq 2$, $n\geq 1$.