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.
State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院数学科学学院）；School of Mathematical Sciences, University of Science and Technology of China（中国科学技术大学数学科学学院）
