Mock modularity of the Md-rank of overpartitions

We investigate the modular properties of a new partition rank, the Md-rank of overpartitions. In fact this is an infinite family of ranks, indexed by the positive integer d, that gives both the Dyson rank of overpartitions and the overpartition M2-rank as special cases. The Md-rank of overpartitions is the holomorphic part of a certain harmonic Maass form of weight 12. We give the exact transformation of this harmonic Maass form along with a few identities for the Md-rank.