Genus and 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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 and . (Hurwitz numbers essentially count irreducible genus 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