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