Combinatorial interpretations of cranks of overpartitions and partitions into distinct odd parts

We give combinatorial interpretations of two residual cranks of overpartitions defined by Bringmann, Lovejoy and Osburn in 2009 analogous to the crank of partitions given by Andrews and Garvan in 1988. As a consequence, we improve upon their definitions and find the true residual cranks of overpartitions. Furthermore, we investigate the combinatorial interpretation of an M₂-crank of partitions without repeated odd parts and explore connections of these statistics with their companion rank counterparts and the tenth order mock theta functions of Ramanujan.