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


Author: Ravi Vakil
Journal: Trans. Amer. Math. Soc. 353 (2001), 4025-4038
MSC (2000): Primary 14H10, 05C30; Secondary 58D29
DOI: https://doi.org/10.1090/S0002-9947-01-02776-3
Published electronically: June 1, 2001
MathSciNet review: 1837218
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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, Jackson and Vainshtein, and extend results of Hurwitz and many others.


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Additional Information

Ravi Vakil
Affiliation: Department of Mathematics, Stanford University, Building 380, MC2125, Stanford, California 94305
Email: vakil@math.stanford.edu

DOI: https://doi.org/10.1090/S0002-9947-01-02776-3
Received by editor(s): December 16, 1998
Published electronically: June 1, 2001
Additional Notes: The author was supported in part by NSF Grant DMS-9970101
Article copyright: © Copyright 2001 American Mathematical Society

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