Hurwitz numbers give a count of genus , degree covers of
with ramification profile
over fixed branch points
. Covers with
automorphism group are counted with weight . The Hurwitz
numbers can be calculated explicitly in terms of the character theory
of the symmetric group. Remarkably, they can also be expressed in
terms of tautological integrals on
by the ELSV formula (see references below).

Ekedahl, Lando, Shapiro, Vainshtein, On Hurwitz numbers
and Hodge integrals. C. R. Acad. Sci. Paris Sér. I Math. 328
(1999), no. 12, 1175-1180.

Hurwitz numbers and intersections on moduli spaces of curves.
Invent. Math. 146 (2001), no. 2, 297-327.

Jeffrey Herschel Giansiracusa
2005-06-27