  In this paper, we show that the Ozsv\'ath-Szab\'o contact invariant
$c^+(\xi)\in HF^+(-Y)$ of a contact 3-manifold $(Y,\xi)$ can be calculated
combinatorially if $Y$ is the boundary of a certain type of plumbing $X$, and
$\xi$ is induced by a Stein structure on $X$. Our technique uses an algorithm
of Ozsv\'ath and Szab\'o to determine the Heegaard-Floer homology of such
3-manifolds. We discuss two important applications of this technique in contact
topology. First, we show that it simplifies the calculation of the
Ozsv\'ath-Stipsicz-Szab\'o obstruction to admitting a planar open book. Then we
define a numerical invariant of contact manifolds that respects a partial
ordering induced by Stein cobordisms. We do a sample calculation showing that
the invariant can get infinitely many distinct values.
