Genus $0$ and $1$ Hurwitz numbers: Recursions, formulas, and graph-theoretic interpretations
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- by Ravi Vakil
- Trans. Amer. Math. Soc. 353 (2001), 4025-4038
- DOI: https://doi.org/10.1090/S0002-9947-01-02776-3
- Published electronically: June 1, 2001
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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.References
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Bibliographic Information
- Ravi Vakil
- Affiliation: Department of Mathematics, Stanford University, Building 380, MC2125, Stanford, California 94305
- MR Author ID: 606760
- ORCID: 0000-0001-8506-270X
- Email: vakil@math.stanford.edu
- 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
- © Copyright 2001 American Mathematical Society
- 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
- MathSciNet review: 1837218