## Pruned Hurwitz numbers

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- by Norman Do and Paul Norbury PDF
- Trans. Amer. Math. Soc.
**370**(2018), 3053-3084 Request permission

## 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
- MR Author ID: 726659
- Email: norm.do@monash.edu
**Paul Norbury**- Affiliation: School of Mathematics and Statistics, The University of Melbourne, Victoria 3010, Australia
- MR Author ID: 361773
- Email: pnorbury@ms.unimelb.edu.au
- 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)
- © Copyright 2017 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**370**(2018), 3053-3084 - MSC (2010): Primary 14N10; Secondary 05A15, 32G15
- DOI: https://doi.org/10.1090/tran/7021
- MathSciNet review: 3766841