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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. 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Zang","submitted_at":"2018-08-13T15:10:55Z","abstract_excerpt":"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. 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