  Let $X = S \times E$ be the product of a K3 surface $S$ and an elliptic curve
$E$. Reduced stable pair invariants of $X$ can be defined via (1) cutting down
the reduced virtual class with incidence conditions or (2) the Behrend function
weighted Euler characteristic of the quotient of the moduli space by the
translation action of $E$. We show that (2) arises naturally as the degree of a
virtual class, and that the invariants (1) and (2) agree. This has applications
to deformation invariance, rationality and a DT/PT correspondence for reduced
invariants of $S \times E$.
