  Hurwitz numbers, which count certain covers of the projective line (or,
equivalently, factorizations of permuations into transpositions), have been
extensively studied for over a century. The Gromov-Witten potential F of a
point, the generating series for Hodge integrals on the moduli space of curves,
has been a central object of study in Gromov-Witten theory. We define a
slightly enriched Gromov-Witten potential G (including integrals involving one
"lambda-class"), and show that after a non-trivial change of variables, G=H in
positive genus, where H is a generating series for Hurwitz numbers. We prove a
conjecture of Goulden and Jackson on higher genus Hurwitz numbers, which turns
out to be an analogue of a genus expansion ansatz of Itzykson and Zuber. As
consequences, we have new combinatorial constraints on F, and a much more
direct proof of the ansatz of Itzykson and Zuber.
  We can produce recursions and explicit formulas for Hurwitz numbers; the
algorithm presented should prove "all" such recursions. Furthermore, there are
many more recursions than previously suspected from geometry (and indeed they
should exist in all genera); as examples we present surprisingly simple new
recursions in genus up to 3 that are geometrically mysterious.
  As we expect this paper also to be of interest to combinatorialists, we have
tried to make it as self-contained as possible, including reviewing some
results and definitions well known in algebraic and symplectic geometry, and
mathematical physics.
