  Noting a curious link between Andrews' even-odd crank and the Stanley rank,
we adopt a combinatorial approach building on the map of conjugation and
continue the study of integer partitions with parts separated by parity. Our
motivation is twofold. First off, we derive results for certain restricted
partitions with even parts below odd parts. These include a Franklin-type
involution proving a parametrized identity that generalizes Andrews' bivariate
generating function, and two families of Andrews--Beck type congruences.
Secondly, we introduce several new subsets of partitions that are stable (i.e.,
invariant under conjugation) and explore their connections with three third
order mock theta functions $\omega(q)$, $\nu(q)$, and $\psi^{(3)}(q)$,
introduced by Ramanujan and Watson.
