The rank of partitions play an important role in the combinatorial interpretations of several Ramanujan's famous congruence formulas. In 2005 and 2008, the $D$-rank and $M_2$-rank of an overpartition were introduced by Lovejoy, respectively. Let $\overline{N}(m,n)$ and $\overline{N2}(m,n)$ denote the number of overpartitions of $n$ with $D$-rank $m$ and $M_2$-rank $m$, respectively. In 2014, Chan and Mao proposed a conjecture on monotonicity properties of $\overline{N}(m,n)$ and $\overline{N2}(m,n)$. In this paper, we prove the Chan-Mao monotonicity conjecture. To be specific, we show that for any integer $m$ and nonnegative integer $n$, $\overline{N2}(m,n)\leq \overline{N2}(m,n+1)$; and for $(m,n)\neq (0,4)$ with $n\neq\, |m| +2$, we have $\overline{N}(m,n)\leq \overline{N}(m,n+1)$. Furthermore, when $m$ increases, we prove that $\overline{N}(m,n)\geq \overline{N}(m+2,n)$ and $\overline{N2}(m,n)\geq \overline{N2}(m+2,n)$ for any $m,n\geq 0$, which is an analogue of Chan and Mao's result for partitions.

中文翻译：

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超分区等级的单调性

分区的等级在几个 Ramanujan 著名的同余公式的组合解释中起着重要作用。在 2005 年和 2008 年，Lovejoy 分别引入了超分区的 $D$-rank 和 $M_2$-rank。让 $\overline{N}(m,n)$ 和 $\overline{N2}(m,n)$ 表示 $n$ 与 $D$-rank $m$ 和 $M_2$-rank 的超分区数分别为 $m$。2014年，Chan和Mao提出了$\overline{N}(m,n)$和$\overline{N2}(m,n)$的单调性猜想。在本文中，我们证明了 Chan-Mao 单调性猜想。具体来说，我们证明对于任何整数 $m$ 和非负整数 $n$， $\overline{N2}(m,n)\leq \overline{N2}(m,n+1)$; 而对于 $(m,n)\neq (0,4)$ 和 $n\neq\, |m| +2$，我们有 $\overline{N}(m,n)\leq \overline{N}(m,n+1)$。此外，当 $m$ 增加时，