Geodesic
Toda Equations and Piecewise Polynomiality for Mixed Double Hurwitz Numbers
Symmetry, integrability and geometry: methods and applications, Tome 12 (2016) Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
@article{SIGMA_2016_12_a39,
     author = {I. P. Goulden and Mathieu Guay-Paquet and Jonathan Novak},
     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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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/

[1] Biane P., “Parking functions of types A and B”, Electron. J. Combin., 9 (2002), 7, 5 pp. | MR

[2] Carrell S. R., “Diagonal solutions to the 2-Toda hierarchy”, Math. Res. Lett., 22 (2015), 439–465, arXiv: 1109.1451 | DOI | MR | Zbl

[3] Diaconis P., Greene C., Applications of Murphy's elements, Stanford University Technical Report No 335, 1989

[4] Do N., Dyer A., Mathews D. V., Topological recursion and a quantum curve for monotone Hurwitz numbers, arXiv: 1408.3992

[5] Ekedahl T., Lando S., Shapiro M., Vainshtein A., “Hurwitz numbers and intersections on moduli spaces of curves”, Invent. Math., 146 (2001), 297–327, arXiv: math.AG/0004096 | DOI | MR | Zbl

[6] Farahat H. K., Higman G., “The centres of symmetric group rings”, Proc. Roy. Soc. London Ser. A, 250 (1959), 212–221 | DOI | MR | Zbl

[7] Goulden I. P., Guay-Paquet M., Novak J., “Monotone Hurwitz numbers in genus zero”, Canad. J. Math., 65 (2013), 1020–1042, arXiv: 1204.2618 | DOI | MR | Zbl

[8] Goulden I. P., Guay-Paquet M., Novak J., “Polynomiality of monotone Hurwitz numbers in higher genera”, Adv. Math., 238 (2013), 1–23, arXiv: 1210.3415 | DOI | MR | Zbl

[9] Goulden I. P., Guay-Paquet M., Novak J., “Monotone Hurwitz numbers and the HCIZ integral”, Ann. Math. Blaise Pascal, 21 (2014), 71–89, arXiv: 1107.1015 | DOI | MR | Zbl

[10] Goulden I. P., Jackson D. M., Vakil R., “Towards the geometry of double Hurwitz numbers”, Adv. Math., 198 (2005), 43–92, arXiv: math.AG/0309440 | DOI | MR | Zbl

[11] Johnson P., “Double Hurwitz numbers via the infinite wedge”, Trans. Amer. Math. Soc., 367 (2015), 6415–6440, arXiv: 1008.3266 | DOI | MR | Zbl

[12] Kazarian M. E., Lando S. K., “An algebro-geometric proof of Witten's conjecture”, J. Amer. Math. Soc., 20 (2007), 1079–1089, arXiv: math.AG/0601760 | DOI | MR | Zbl

[13] Matsumoto S., Novak J., “Jucys–Murphy elements and unitary matrix integrals”, Int. Math. Res. Not., 2013 (2013), 362–397, arXiv: 0905.1992 | DOI | MR | Zbl

[14] Novak J., “Vicious walkers and random contraction matrices”, Int. Math. Res. Not., 2009 (2009), 3310–3327, arXiv: 0705.0984 | DOI | MR | Zbl

[15] Okounkov A., “Toda equations for Hurwitz numbers”, Math. Res. Lett., 7 (2000), 447–453, arXiv: math.AG/0004128 | DOI | MR | Zbl

[16] Okounkov A., Pandharipande R., “Gromov–Witten theory, Hurwitz theory, and completed cycles”, Ann. of Math., 163 (2006), 517–560, arXiv: math.AG/0204305 | DOI | MR | Zbl

[17] Okounkov A., Pandharipande R., “The equivariant Gromov–Witten theory of $P^1$”, Ann. of Math., 163 (2006), 561–605, arXiv: math.AG/0207233 | DOI | MR | Zbl

[18] Olshanski G., “Plancherel averages: remarks on a paper by Stanley”, Electron. J. Combin., 17 (2010), 43, 16 pp., arXiv: 0905.1304 | MR

[19] Orlov A. Yu., Shcherbin D. M., “Hypergeometric solutions of soliton equations”, Theoret. and Math. Phys., 128 (2001), 906–926 | DOI | MR | Zbl

[20] Pandharipande R., “The Toda equations and the Gromov–Witten theory of the Riemann sphere”, Lett. Math. Phys., 53 (2000), 59–74, arXiv: math.AG/9912166 | DOI | MR

[21] Shadrin S., Spitz L., Zvonkine D., “On double Hurwitz numbers with completed cycles”, J. London Math. Soc., 86 (2012), 407–432, arXiv: 1103.3120 | DOI | MR | Zbl

[22] Stanley R. P., “Parking functions and noncrossing partitions”, Electron. J. Combin., 4 (1997), 20, 14 pp. | MR