$M_2$-Ranks of overpartitions modulo 4 and 8

Abstract

An overpartition is a partition in which the first occurrence of a number may be overlined. For an overpartition $\lambda$, let $\ell (\lambda )$ denote the largest part of $\lambda$, and let $n(\lambda )$ denote its number of parts. Then the $M_2$-rank of an overpartition is defined as

$$\begin{aligned} M_2\text {-rank}(\lambda ):=\left\lceil \frac{\ell (\lambda )}{2}\right\rceil -n(\lambda )+n(\lambda _0)-\chi (\lambda ), \end{aligned}$$

where $\chi (\lambda )=1$ if $\ell (\lambda )$ is odd and non-overlined and $\chi (\lambda )=0$, otherwise. In this paper, we study the $M_2$-rank differences of overpartitions modulo 4 and 8. Especially, we obtain some relations between the generating functions of the $M_2$-rank differences modulo 4 and 8 and the second order mock theta functions. Furthermore, we deduce some inequalities on $M_2$-ranks of overpartitions.