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arXiv · 2308.04678

Inequalities for the $k$-Regular Overpartitions

Yi Peng·Helen W. J. Zhang·Ying Zhong
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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 $2\leq k\leq9$.

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Yi Peng, Helen W. J. Zhang, Ying Zhong. 2023-08-09. Inequalities for the $k$-Regular Overpartitions. https://arxiv.org/abs/2308.04678

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