Hurwitz numbers

Hurwitz numbers give a count of genus $g$, degree $d$ covers of ${\mathbf{P}}^1$ with ramification profile ${\mu }_1,\dots {\mu }_b$ over fixed branch points $p_1,\dots p_b$. Covers with automorphism group $G$ are counted with weight $|G|$. 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 ${\overline{\mathcal{M}}}_{g,n}$ 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.

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