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


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:
  • Paul Norbury
  • Affiliation: School of Mathematics and Statistics, The University of Melbourne, Victoria 3010, Australia
  • MR Author ID: 361773
  • Email:
  • 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:
  • MathSciNet review: 3766841