Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 040, 10 pages      arXiv:1307.2137      https://doi.org/10.3842/SIGMA.2016.040
Contribution to the Special Issue on Asymptotics and Universality in Random Matrices, Random Growth Processes, Integrable Systems and Statistical Physics in honor of Percy Deift and Craig Tracy

Toda Equations and Piecewise Polynomiality for Mixed Double Hurwitz Numbers

I.P. Goulden a, Mathieu Guay-Paquet b and Jonathan Novak c
a) Department of Combinatorics and Optimization, University of Waterloo, 200 University Ave. W., Waterloo, ON, N2L 3G1 Canada
b) Département de mathématiques, Université du Québec à Montréal, C.P. 8888, succ. Centre-ville, Montréal, Québec, H3C 3P8 Canada
c) Department of Mathematics, University of California San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0404 USA

Received February 02, 2016, in final form April 13, 2016; Published online April 20, 2016

Abstract
This article introduces mixed double Hurwitz numbers, which interpolate combinatorially between the classical double Hurwitz numbers studied by Okounkov and the monotone double Hurwitz numbers introduced recently by Goulden, Guay-Paquet and Novak. Generalizing a result of Okounkov, we prove that a certain generating series for the mixed double Hurwitz numbers solves the 2-Toda hierarchy of partial differential equations. We also prove that the mixed double Hurwitz numbers are piecewise polynomial, thereby generalizing a result of Goulden, Jackson and Vakil.

Key words: Hurwitz numbers; Toda lattice.

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