Monotonicity properties for ranks of overpartitions

TL;DR

This paper proves a conjecture about the monotonicity of overpartition ranks, showing how their counts change with parameters and establishing inequalities that extend known results for partitions.

Contribution

It confirms the Chan-Mao monotonicity conjecture for overpartition ranks and extends related inequalities to overpartitions, enriching the combinatorial understanding of these objects.

Findings

Proved that ar{N2}(m,n) ; ar{N2}(m,n+1) for all m,n.

Established ar{N}(m,n) ; ar{N}(m,n+1) for most m,n.

Showed inequalities relating ar{N}(m,n) and ar{N}(m+2,n) for all m,n.

Abstract

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.