Recursions, formulas, and graph-theoretic interpretations of ramified coverings of the sphere by surfaces of genus 0 and 1

Ravi Vakil

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arXiv · math/9812105

Recursions, formulas, and graph-theoretic interpretations of ramified coverings of the sphere by surfaces of genus 0 and 1

Ravi Vakil

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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 and Jackson, and extend results of Hurwitz and many others.

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Ravi Vakil. 1998-12-17. Recursions, formulas, and graph-theoretic interpretations of ramified coverings of the sphere by surfaces of genus 0 and 1. https://arxiv.org/abs/math/9812105

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