$$M_2$$-Ranks of overpartitions modulo 4 and 8
2020
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.
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