Strict Log-Subadditivity for Overpartition Rank

Bessenrodt and Ono initially found the strict log-subadditivity of partition function $p(n)$, that is, $p(a+b)<p(a)p(b)$ for $a,b>1$ and $a+b>9$. Many other important statistics of partitions are proved to enjoy similar properties. Lovejoy introduced the overpartition rank as an analog of Dyson’s rank for partitions from the $q$-series perspective. Let $\bar{N}(a,c,n)$ denote the number of overpartitions with rank congruent to $a$ modulo $c$. Ciolan computed the asymptotic formula of $\bar{N}(a,c,n)$ and showed that $\bar{N}(a,c,n)>\bar{N}(b,c,n)$ for $c\ge 7$ and $n$ large enough. In this paper, we derive an upper bound and a lower bound of $\bar{N}(a,c,n)$ for each $c\ge 3$ by using the asymptotics of Ciolan. Consequently, we establish the strict log-subadditivity of $\bar{N}(a,c,n)$ analogous to the partition function $p(n)$.