  Let $Y$ be a smooth projective threefold and let $f:Y\to X$ be a birational
map with $Rf_*\mathcal{O}_Y=\mathcal{O}_X$. When $Y$ is Calabi-Yau,
Bryan-Steinberg defined enumerative invariants associated to such maps called
$f$-relative stable (or Bryan-Steinberg) invariants. When $X$ has Gorenstein
singularities and $f$ has relative dimension one, they compared these
invariants to the Donaldson-Thomas, or equivalently the Pandharipande-Thomas
invariants of $Y$.
  We define Bryan-Steinberg invariants for maps $f$ as above without assuming
that $Y$ is Calabi-Yau. For $X$ with Gorenstein and rational singularities, $f$
of relative dimension one, and for insertions from $X$ and arbitrary descendant
levels, we conjecture a relation between the generating functions of
Bryan-Steinberg and Pandharipande-Thomas invariants of $Y$. We check the
conjecture for the contraction $f: Y\to X$ of a rational curve $C$ with normal
bundle $N_{C/Y}\cong \mathcal{O}_C(-1)^{\oplus 2}$ using degeneration and
localization techniques to reduce to a Calabi-Yau situation, which we then
treat using Joyce's motivic Hall algebra.
