Explicit formulae for one-part double Hurwitz numbers with completed 3-cycles

  • Viet Anh Nguyen


We prove two explicit formulae for one-part double Hurwitz numbers with completed 3-cycles. We define “combinatorial Hodge integrals” from these numbers in the spirit of the celebrated ELSV formula. The obtained results imply some explicit formulae and properties of the combinatorial Hodge integrals.


Hurwitz numbers Symmetric groups Symmetric functions 



This article is a part of my Ph.D. thesis that I am preparing under the supervision of Mattia Cafasso and Vladimir Roubtsov at LAREMA, UMR CNRS 6093. I am grateful to my supervisors for continuous support. I am also grateful to Bertrand Eynard and Vincent Rivasseau for having guided my first steps in research with extreme care during my research internship. This experience played a key role in my decision to continue the hard path of research. My Ph.D. study is funded by the French ministerial scholarship “Allocations Spécifiques Polytechniciens”. My research is also partially supported by LAREMA and the Nouvelle Équipe “Topologie algébrique et Physique Mathématique” of the Pays de la Loire region.


  1. 1.
    Alexandrov, A.: Enumerative geometry, tau-functions and Heisenberg–Virasoro algebra. Comm. Math. Phys. 338(1), 195–249 (2015)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Cavalieri, R., Johnson, P., Markwig, H.: Tropical Hurwitz numbers. J. Algebraic Combin. 32(2), 241–265 (2010)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Ekedahl, T., Lando, S., Shapiro, M., Vainshtein, A.: Hurwitz numbers and intersections on moduli spaces of curves. Invent. Math. 146(2), 297–327 (2001)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Eynard, B., Mulase, M., Safnuk, B.: The Laplace transform of the cut-and-join equation and the Bouchard-Mariño conjecture on Hurwitz numbers. Publ. Res. Inst. Math. Sci. 47(2), 629–670 (2011)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Faber, C., Pandharipande, R.: Hodge integrals, partition matrices, and the \(\lambda _g\) conjecture. Ann. Math. (2) 157(1), 97–124 (2003)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Goulden, I.P., Jackson, D.M., Vakil, R.: Towards the geometry of double Hurwitz numbers. Adv. Math. 198(1), 43–92 (2005)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Johnson, P., Pandharipande, R., Tseng, H.-H.: Abelian Hurwitz-Hodge integrals. Michigan Math. J. 60(1), 171–198, 04 (2011)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Kazarian, M.E., Lando, S.K.: Combinatorial solutions to integrable hierarchies. Russian Math. Surveys 70(3), 453 (2015)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Kerov, S., Olshanski, G.: Polynomial functions on the set of Young diagrams. C. R. Acad. Sci. Paris Sér. I Math. 319(2), 121–126 (1994)MathSciNetMATHGoogle Scholar
  10. 10.
    Lando, S.K., Zvonkin, A.K.: Graphs on Surfaces and Their Applications. Springer, Berlin (2004)CrossRefMATHGoogle Scholar
  11. 11.
    Okounkov, A.: Toda equations for Hurwitz numbers. Math. Res. Lett. 7(4), 447–453 (2000)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Okounkov, A., Pandharipande, R.: Gromov-Witten theory, Hurwitz theory, and completed cycles. Ann. Math. (2) 163(2), 517–560 (2006)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Shadrin, S., Spitz, L., Zvonkine, D.: On double Hurwitz numbers with completed cycles. J. Lond. Math. Soc. (2) 86(2), 407–432 (2012)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Shadrin, S., Spitz, L., Zvonkine, D.: Equivalence of ELSV and Bouchard–Mariño conjectures for \(r\)-spin Hurwitz numbers. Math. Ann. 361(3–4), 611–645 (2015)MathSciNetCrossRefMATHGoogle Scholar

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Authors and Affiliations

  1. 1.LAREMA UMR CNRS 6093Université d’AngersAngersFrance

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