Inequalities for the $k$-Regular Overpartitions

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 Turán 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 $k\ge 2$, 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 $a\ge b\ge 1$ and $a+b\ge k$. Furthermore, we investigate the log-concavity and the satisfaction of the third-order Turán inequality for ${\overline{p}}_k(n)$, where $2\le k\le 9$.

28 pages