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Pruned Hurwitz numbers


Authors: Norman Do and Paul Norbury
Journal: Trans. Amer. Math. Soc. 370 (2018), 3053-3084
MSC (2010): Primary 14N10; Secondary 05A15, 32G15
DOI: https://doi.org/10.1090/tran/7021
Published electronically: November 16, 2017
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Abstract: Simple Hurwitz numbers count branched covers of the Riemann sphere and are well-studied in the literature. We define a new enumeration that restricts the count to branched covers satisfying an additional constraint. The resulting pruned Hurwitz numbers determine their simple counterparts, but have the advantage of satisfying simpler recursion relations and obeying simpler formulae. As an application of pruned Hurwitz numbers, we obtain a new proof of the Witten-Kontsevich theorem. Furthermore, we apply the idea of defining useful restricted enumerations to orbifold Hurwitz numbers and Belyi Hurwitz numbers.


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Additional Information

Norman Do
Affiliation: School of Mathematical Sciences, Monash University, Victoria 3800, Australia
Email: norm.do@monash.edu

Paul Norbury
Affiliation: School of Mathematics and Statistics, The University of Melbourne, Victoria 3010, Australia
Email: pnorbury@ms.unimelb.edu.au

DOI: https://doi.org/10.1090/tran/7021
Keywords: Hurwitz numbers, fatgraphs, topological recursion
Received by editor(s): November 16, 2015
Received by editor(s) in revised form: July 14, 2016
Published electronically: November 16, 2017
Additional Notes: The authors were partially supported by the Australian Research Council grants DE130100650 (ND) and DP1094328 (PN)
Article copyright: © Copyright 2017 American Mathematical Society

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