Strict Log-Subadditivity for Overpartition Rank

TL;DR

This paper proves the strict log-subadditivity property for the overpartition rank statistic, extending known results from the partition function to a broader class of partition-related statistics.

Contribution

It establishes the strict log-subadditivity of the overpartition rank, a novel property for this statistic, using asymptotic bounds derived from prior asymptotic formulas.

Findings

Established upper and lower bounds for overpartition rank counts.

Proved strict log-subadditivity of overpartition rank for all relevant parameters.

Extended the concept of log-subadditivity from partition functions to overpartition ranks.

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 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 $\overline{N}(a,c,n)$ 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 $c\geq7$ and $n$ large enough. In this paper, we derive an upper bound and a lower bound of $\overline{N}(a,c,n)$ for each $c\geq3$ by using the asymptotics of Ciolan. Consequently, we establish the strict log-subadditivity of $\overline{N}(a,c,n)$ analogous to the partition function $p(n)$.