  We construct curve counting invariants for a Calabi-Yau threefold $Y$
equipped with a dominant birational morphism $\pi:Y \to X$. Our invariants
generalize the stable pair invariants of Pandharipande and Thomas which occur
for the case when $\pi:Y\to Y$ is the identity. Our main result is a PT/DT-type
formula relating the partition function of our invariants to the
Donaldson-Thomas partition function in the case when $Y$ is a crepant
resolution of $X$, the coarse space of a Calabi-Yau orbifold $\mathcal{X}$
satisfying the hard Lefschetz condition. In this case, our partition function
is equal to the Pandharipande-Thomas partition function of the orbifold
$\mathcal{X}$. Our methods include defining a new notion of stability for
sheaves which depends on the morphism $\pi $. Our notion generalizes slope
stability which is recovered in the case where $\pi $ is the identity on $Y$.
Our proof is a generalization of Bridgeland's proof of the PT/DT correspondence
via the Hall algebra and Joyce's integration map.
