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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Genus $0$ and $1$ Hurwitz numbers: Recursions, formulas, and graph-theoretic interpretations
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by Ravi Vakil PDF
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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Additional 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