Strict Log-Subadditivity for Overpartition Rank

Abstract

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 partition statistics 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 \({\overline{N}}(a,c,n)\) denote the number of overpartitions with rank congruent to a modulo c. Ciolan computed the asymptotic formula of \({\overline{N}}(a,c,n)\) and showed that \({\overline{N}}(a, c, n) > {\overline{N}}(b, c, n)\) for \(0\le a<b\le \lfloor \frac{c}{2}\rfloor \) and n large enough if \(c\ge 7\). In this paper, we derive an upper bound and a lower bound of \({\overline{N}}(a,c,n)\) for each \(c\ge 3\) by using the asymptotics due to Ciolan. Consequently, we establish the strict log-subadditivity of \({\overline{N}}(a,c,n)\) analogous to the partition function p(n).