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Moscow Mathematical Journal

Volume 17, Issue 4, October–December 2017  pp. 601–633.

Classical Hurwitz Numbers and Related Combinatorics

Authors:  Boris Dubrovin (1), Di Yang (2), and Don Zagier (2)
Author institution:(1) SISSA, via Bonomea 265, Trieste 34136, Italy
(2) Max-Planck-Institut für Mathematik, Vivatsgasse 7, Bonn 53111, Germany


We give a polynomial-time algorithm of computing the classical Hurwitz numbers Hg,d, which were defined by Hurwitz 125 years ago. We show that the generating series of Hg,d for any fixed g ≥ 2 lives in a certain subring of the ring of formal power series that we call the Lambert ring. We then define some analogous numbers appearing in enumerations of graphs, ribbon graphs, and in the intersection theory on moduli spaces of algebraic curves, such that their generating series belong to the same Lambert ring. Several asymptotics of these numbers (for large g or for large d) are obtained.

2010 Math. Subj. Class. Primary: 14N10; Secondary: 16T30, 53D45, 05A15.

Keywords: Hurwitz numbers, Lambert ring, Pandharipande’s equation, enumerative geometry.

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