Inequalities for the $k$-Regular Overpartitions

TL;DR

This paper proves that the overpartition function $ar{p}_k(n)$ is strictly log-subadditive for all $k\,\geq 2$, and explores its log-concavity and Turán inequalities for specific $k$ values.

Contribution

It provides a combinatorial proof of strict log-subadditivity of $ar{p}_k(n)$ and investigates its log-concavity and Turán inequalities for $k$ between 2 and 9.

Findings

$ar{p}_k(a)ar{p}_k(b) > ar{p}_k(a+b)$ for $a \,\geq b \,\geq 1$ and $a+b \,\geq k$

Evidence of log-concavity and Turán inequalities for $ar{p}_k(n)$ when $2 \,\leq k \,\leq 9$

Abstract

Bessenrodt and Ono, Chen, Wang and Jia, DeSalvo and Pak were the first to discover the log-subadditivity, log-concavity, and the third-order Tur\'{a}n inequality of partition function, respectively. Many other important partition statistics are proved to enjoy similar properties. This paper focuses on the partition function $\overline{p}_k(n)$, which counts the number of overpartitions of $n$ with no parts divisible by $k$. We provide a combinatorial proof to establish that for any $k\geq2$, the partition function $\overline{p}_k(n)$ exhibits strict log-subadditivity. Specifically, we show that $\overline{p}_k(a)\overline{p}_k(b)>\overline{p}_k(a+b)$ for integers $a\geq b\geq1$ and $a+b\geq k$. Furthermore, we investigate the log-concavity and the satisfaction of the third-order Tur\'{a}n inequality for $\overline{p}_k(n)$, where $2\leq k\leq9$.