Genus 0 and 1 Hurwitz numbers: Recursions, formulas, and graph-theoretic interpretations

Vakil, Ravi

doi:10.1090/S0002-9947-01-02776-3

Genus $0$ and $1$ Hurwitz numbers: Recursions, formulas, and graph-theoretic interpretations
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Trans. Amer. Math. Soc. 353 (2001), 4025-4038 Request permission

Abstract:

We derive a closed-form expression for all genus 1 Hurwitz numbers, and give a simple new graph-theoretic interpretation of Hurwitz numbers in genus $0$ and $1$. (Hurwitz numbers essentially count irreducible genus $g$ covers of the sphere, with arbitrary specified branching over one point, simple branching over other specified points, and no other branching. The problem is equivalent to counting transitive factorisations of permutations into transpositions.) These results prove a conjecture of Goulden, Jackson and Vainshtein, and extend results of Hurwitz and many others.

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