  Hurwitz numbers count ramified covers of a Riemann surface with prescribed
monodromy. As such, they are purely combinatorial objects. Tautological
classes, on the other hand, are distinguished classes in the intersection ring
of the moduli spaces of Riemann surfaces of a given genus, and are thus
``geometric.'' Localization computations in Gromov-Witten theory provide
non-obvious relations between the two. This paper makes one such computation,
and shows how it leads to a ``master'' relation (Theorem 0.1) that reduces the
ratios of certain interesting tautological classes to the pure combinatorics of
Hurwitz numbers. As a corollary, we obtain a purely combinatorial proof of a
theorem of Bryan and Pandharipande, expressing in generating function form
classical computations by Faber/Looijenga (Theorem 0.2).
