  Let $E$ be a rank 2, degree $d$ vector bundle over a genus $g$ curve $C$. The
loci of stable pairs on $E$ in class $2[C]$ fixed by the scaling action are
expressed as products of $\Quot$ schemes. Using virtual localization, the
stable pairs invariants of $E$ are related to the virtual intersection theory
of $\Quot E$. The latter theory is extensively discussed for an $E$ of
arbitrary rank; the tautological ring of $\Quot E$ is defined and is computed
on the locus parameterizing rank one subsheaves. In case $E$ has rank 2, $d$
and $g$ have opposite parity, and $E$ is sufficiently generic, it is known that
$E$ has exactly $2^g$ line subbundles of maximal degree. Doubling the zero
section along such a subbundle gives a curve in the total space of $E$ in class
$2[C]$. We relate this count of maximal subbundles with stable
pairs/Donaldson-Thomas theory on the total space of $E$. This endows the
residue invariants of $E$ with enumerative significance: they actually
\emph{count} curves in $E$.
