Monotonicity properties for ranks of overpartitions

The rank of partitions plays an important role in the combinatorial interpretations of several Ramanujan's famous congruence formulas. In 2005 and 2008, the D-rank and M₂-rank of an overpartition were introduced by Lovejoy, respectively. Let N‾(m,n) and N2‾(m,n) denote the number of overpartitions of n with D-rank m and M₂-rank m, respectively. In 2014, Chan and Mao proposed a conjecture on monotonicity properties of N‾(m,n) and 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, N2‾(m,n)≤N2‾(m,n+1); and for (m,n)≠(0,4) with n≠|m|+2, we have N‾(m,n)≤N‾(m,n+1). Furthermore, when m increases, we prove that N‾(m,n)≥N‾(m+2,n) and N2‾(m,n)≥N2‾(m+2,n) for any m,n≥0, which is an analogue of Chan and Mao's result for partitions.