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On the Convergence of Monotone Hurwitz Generating Functions

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Abstract

Monotone Hurwitz numbers were introduced by the authors as a combinatorially natural desymmetrization of the Hurwitz numbers studied in enumerative algebraic geometry. Over the course of several papers, we developed the structural theory of monotone Hurwitz numbers and demonstrated that it is in many ways parallel to that of their classical counterparts. In this note, we identify an important difference between the monotone and classical worlds: fixed-genus generating functions for monotone double Hurwitz numbers are absolutely summable, whereas those for classical double Hurwitz numbers are not. This property is crucial for applications of monotone Hurwitz theory in analysis. We quantify the growth rate of monotone Hurwitz numbers in fixed genus by giving universal upper and lower bounds on the radii of convergence of their generating functions.

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Authors and Affiliations

  1. Department of Combinatorics and Optimization, University of Waterloo, 200 University Ave. W., Waterloo, ON, N2L 3G1, Canada

    I. P. Goulden

  2. LACIM, Université du Québec à Montréal, C.P. 8888, succ. Centre-ville, Montréal, Québec, H3C 3P8, Canada

    Mathieu Guay-Paquet

  3. Department of Mathematics, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA, 92093-0404, USA

    Jonathan Novak

Authors
  1. I. P. Goulden
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  2. Mathieu Guay-Paquet
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  3. Jonathan Novak
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Correspondence to I. P. Goulden.

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Goulden, I.P., Guay-Paquet, M. & Novak, J. On the Convergence of Monotone Hurwitz Generating Functions. Ann. Comb. 21, 73–81 (2017). https://doi.org/10.1007/s00026-017-0341-5

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  • DOI: https://doi.org/10.1007/s00026-017-0341-5

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