Geodesic
Toda Equations and Piecewise Polynomiality for Mixed Double Hurwitz Numbers
Symmetry, integrability and geometry: methods and applications, Tome 12 (2016)

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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.
Keywords: Hurwitz numbers; Toda lattice. 
I. P. Goulden; Mathieu Guay-Paquet; Jonathan Novak. Toda Equations and Piecewise Polynomiality for Mixed Double Hurwitz Numbers. Symmetry, integrability and geometry: methods and applications, Tome 12 (2016). http://geodesic.mathdoc.fr/item/SIGMA_2016_12_a39/
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     title = {Toda {Equations} and {Piecewise} {Polynomiality} for {Mixed} {Double} {Hurwitz} {Numbers}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2016},
     volume = {12},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2016_12_a39/}
}
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