Journal of Algebraic Combinatorics

, Volume 48, Issue 2, pp 307–323

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

• Viet Anh Nguyen
Article

## Abstract

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.

## Keywords

Hurwitz numbers Symmetric groups Symmetric functions

## Notes

### Acknowledgements

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.

## References

1. 1.
Alexandrov, A.: Enumerative geometry, tau-functions and Heisenberg–Virasoro algebra. Comm. Math. Phys. 338(1), 195–249 (2015)
2. 2.
Cavalieri, R., Johnson, P., Markwig, H.: Tropical Hurwitz numbers. J. Algebraic Combin. 32(2), 241–265 (2010)
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)
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)
5. 5.
Faber, C., Pandharipande, R.: Hodge integrals, partition matrices, and the $$\lambda _g$$ conjecture. Ann. Math. (2) 157(1), 97–124 (2003)
6. 6.
Goulden, I.P., Jackson, D.M., Vakil, R.: Towards the geometry of double Hurwitz numbers. Adv. Math. 198(1), 43–92 (2005)
7. 7.
Johnson, P., Pandharipande, R., Tseng, H.-H.: Abelian Hurwitz-Hodge integrals. Michigan Math. J. 60(1), 171–198, 04 (2011)
8. 8.
Kazarian, M.E., Lando, S.K.: Combinatorial solutions to integrable hierarchies. Russian Math. Surveys 70(3), 453 (2015)
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)
10. 10.
Lando, S.K., Zvonkin, A.K.: Graphs on Surfaces and Their Applications. Springer, Berlin (2004)
11. 11.
Okounkov, A.: Toda equations for Hurwitz numbers. Math. Res. Lett. 7(4), 447–453 (2000)
12. 12.
Okounkov, A., Pandharipande, R.: Gromov-Witten theory, Hurwitz theory, and completed cycles. Ann. Math. (2) 163(2), 517–560 (2006)
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)
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)