Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Monotonicity properties for ranks of overpartitions   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.

4
The categories of this paper are: cond-mat
24
The categories of this paper are: dg-ga
26
The categories of this paper are: bayes-an
37
The categories of this paper are: supr-con
36
The categories of this paper are: cmp-lg
10
The categories of this paper are: q-bio
21
The categories of this paper are: adap-org
19
The categories of this paper are: eess
28
The categories of this paper are: alg-geom
27
The categories of this paper are: comp-gas
23
The categories of this paper are: solv-int
11
The categories of this paper are: hep-lat
25
The categories of this paper are: patt-sol
20
The categories of this paper are: acc-phys
5
The categories of this paper are: gr-qc
33
The categories of this paper are: plasm-ph