Rank and conjugation for a second Frobenius representation of an overpartition (Q938215)

An overpartition \(\lambda\) of the positive integer \(n\) is a partition of \(n\) in which the first occurrence of a number may be overlined. Let \(\mathcal \ell(\lambda)\) and \(n(\lambda)\) denote the largest part and the number of parts of \(\lambda\), respectively, and let \(\lambda_0\) be the partition consisting of the non-overlined odd parts of \(\lambda\). The \(M_2\)-rank of an overpartition \(\lambda\) is defined by \(\lceil\frac{\mathcal \ell(\lambda)}{2}\rceil- n(\lambda)+n(\lambda_o)-\chi(\lambda)\), where \(\chi(\lambda)=1\) if the largest part in \(\lambda\) is odd and non-overlined and \(\chi(\lambda)=0\) otherwise. The first principal object of this paper is to establish some generating functions for the \(M_2\)-rank in terms of the \(q\)-series notation. The author shows that there exists a combinatorial involution on overpartitions that reverses the sign of the \(M_2\)-rank. He calls it \(2F\)-conjugation. It turns out that \(2F\)-conjugations are different from classical Ferrers diagram conjugations of overpartitions. The last goal of this paper is to give some examples of how to use \(q\)-series identities to derive identities for overpartitions which are \(2F\)-self-conjugate.