The Unimodality of the Crank on Overpartitions

Abstract

Let $N(m,n)$ denote the number of partitions of $n$ with rank $m$, and let $M(m,n)$ denote the number of partitions of $n$ with crank $m$. Chan and Mao proved that for any nonnegative integers $m$ and $n$, $N(m,n)\geq N(m+2,n)$ and for any nonnegative integers $m$ and $n$ such that $n\geq12$, $n\neq m+2$, $N(m,n)\geq N(m,n-1)$. Recently, Ji and Zang showed that for $n\geq 44$ and $1\leq m\leq n-1$, $M(m-1,n)\geq M(m,n)$ and for $n\geq 14$ and $0\leq m\leq n-2$, $M(m,n)\geq M(m,n-1)$. In this paper, we analogue the result of Ji and Zang to overpartitions. Note that Bringmann, Lovejoy and Osburn introduced two type of cranks on overpartitions, namely the first residue crank and the second residue crank. Consequently, for the first residue crank $\overline{M}(m,n)$, we show that $\overline{M}(m-1,n)\geq \overline{M}(m,n)$ for $m\geq 1$ and $n\geq 3$ and $\overline{M}(m,n)\geq \overline{M}(m,n+1)$ for $m\geq 0$ and $n\geq 1$. For the second residue crank $\overline{M2}(m,n)$, we show that $\overline{M2}(m-1,n)\geq \overline{M2}(m,n)$ for $m\geq 1$ and $n\geq 0$ and $\overline{M2}(m,n)\geq \overline{M2}(m,n+1)$ for $m\geq 0$ and $n\geq 1$. Moreover, let $M_k(m,n)$ denote the number of $k$-colored partitions of $n$ with $k$-crank $m$, which was defined by Fu and Tang. They conjectured that when $k\geq 2$, $M_k(m-1,n)\geq M_k(m,n)$ except for $k=2$ and $n=1$. With the aid of the inequality $\overline{M}(m-1,n)\geq \overline{M}(m,n)$ for $m\geq 1$ and $n\geq 3$, we confirm this conjecture.