  We prove a formula for the Hodge numbers of square-free divisors of
Calabi-Yau threefold hypersurfaces in toric varieties. Euclidean branes
wrapping divisors affect the vacuum structure of Calabi-Yau compactifications
of type IIB string theory, M-theory, and F-theory. Determining the
nonperturbative couplings due to Euclidean branes on a divisor $D$ requires
counting fermion zero modes, which depend on the Hodge numbers
$h^i({\cal{O}}_D)$. Suppose that $X$ is a smooth Calabi-Yau threefold
hypersurface in a toric variety $V$, and let $D$ be the restriction to $X$ of a
square-free divisor of $V$. We give a formula for $h^i({\cal{O}}_D)$ in terms
of combinatorial data. Moreover, we construct a CW complex $\mathscr{P}_D$ such
that $h^i({\cal{O}}_D)=h_i(\mathscr{P}_D)$. We describe an efficient algorithm
that makes possible for the first time the computation of sheaf cohomology for
such divisors at large $h^{1,1}$. As an illustration we compute the Hodge
numbers of a class of divisors in a threefold with $h^{1,1}=491$. Our results
are a step toward a systematic computation of Euclidean brane superpotentials
in Calabi-Yau hypersurfaces.
