Graphs and Combinatorics

, Volume 35, Issue 3, pp 729–766 | Cite as

A Monodromy Graph Approach to the Piecewise Polynomiality of Simple, Monotone and Grothendieck Dessins d’enfants Double Hurwitz Numbers

  • Marvin Anas HahnEmail author
Original Paper


Hurwitz numbers count genus g, degree d covers of the complex projective line with fixed branched locus and fixed ramification data. An equivalent description is given by factorisations in the symmetric group. Simple double Hurwitz numbers are a class of Hurwitz-type counts of specific interest. In recent years a related counting problem in the context of random matrix theory was introduced as so-called monotone Hurwitz numbers. These can be viewed as a desymmetrised version of the Hurwitz-problem. A combinatorial interpolation between simple and monotone double Hurwitz numbers was introduced as mixed double Hurwitz numbers and it was proved that these objects are piecewise polynomial in a certain sense. Moreover, the notion of strictly monotone Hurwitz numbers has risen in interest as it is equivalent to a certain Grothendieck dessins d’enfant count. In this paper, we introduce a combinatorial interpolation between simple, monotone and strictly monotone double Hurwitz numbers as triply interpolated Hurwitz numbers. Our aim is twofold: using a connection between triply interpolated Hurwitz numbers and tropical covers in terms of so-called monodromy graphs, we give algorithms to compute the polynomials for triply interpolated Hurwitz numbers in all genera using Erhart theory. We further use this approach to study the wall-crossing behaviour of triply interpolated Hurwitz numbers in genus 0 in terms of related Hurwitz-type counts. All those results specialise to the extremal cases of simple, monotone and Grothendieck dessins d’enfants Hurwitz numbers.


Hurwitz numbers Monodromy graphs Wall-crossing 

Mathematics Subject Classification

14T05 14N10 05A15 



I am indebted to my advisor Hannah Markwig for many helpful suggestions, her guidance and extensive proof-reading throughout the preparation of this paper. Moreover, the author thanks Maksim Karev, Reinier Kramer and Danilo Lewanski for helpful comments and discusions. The author gratefully acknowledges partial support by DFG SFB-TRR 195 “Symbolic tool in mathematics and their applications”, project A 14 “Random matrices and Hurwitz numbers” (INST 248/238-1).


  1. 1.
    Alexandrov, A., Chapuy, G, Eynard, B., Harnad, J.: Weighted Hurwitz numbers and topological recursion: an overview (2016). arXiv:1610.09408
  2. 2.
    Alexandrov, A., Lewanski, D., Shadrin, S.: Ramifications of Hurwitz theory, KP integrability and quantum curves. J. High Energy Phys. 2016(5), 124 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Baldoni, V., Berline, N., De Loera, J., Köppe, M., Vergne, M.: Three Ehrhart quasi-polynomials (2014). arXiv:1410.8632
  4. 4.
    Bertrand, B., Brugallé, E., Mikhalkin, G.: Tropical open hurwitz numbers. Rendiconti del Seminario Matematico della Universitá di Padova 125, 157–171 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chekhov, L., Eynard, B.: Hermitian matrix model free energy: Feynman graph technique for all genera. J. High Energy Phys. 2006(03), 014 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cavalieri, R., Johnson, P., Markwig, H.: Tropical Hurwitz numbers. J. Algebra Combin. 32(2), 241–265 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cavalieri, R., Johnson, P., Markwig, H.: Wall crossings for double Hurwitz numbers. Adv. Math. 228(4), 1894–1937 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cavalieri, R., Miles, E.: Riemann Surfaces and Algebraic Curves. London Mathematical Society Student Texts, vol. 48. Cambridge University Press, Cambridge (2016). A first course in Hurwitz theoryCrossRefzbMATHGoogle Scholar
  9. 9.
    Do, N., Dyer, A., Mathews, D.V.: Topological recursion and a quantum curve for monotone Hurwitz numbers (2014). arXiv:1408.3992
  10. 10.
    Do, N., Karev, M.: Monotone orbifold Hurwitz numbers. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 446(Kombinatorika i Teoriya Grafov. V), 40–69 (2016)Google Scholar
  11. 11.
    Dunin-Barkowski, P., Lewanski, D., Popolitov, A., Shadrin, S.: Polynomiality of orbifold Hurwitz numbers, spectral curve, and a new proof of the Johnson–Pandharipande–Tseng formula. J. Lond. Math. Soc. II. Ser. 92(3), 547–565 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Do, N., Manescu, D.: Quantum curves for the enumeration of ribbon graphs and hypermaps (2013). arXiv:1312.6869
  13. 13.
    Dumitrescu, O., Mulase, M., Safnuk, B., Sorkin, A.: The spectral curve of the eynard-orantin recursion via the laplace transform. Contemp. Math 593, 263–315 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Ekedahl, T., Lando, S., Shapiro, M., Vainshtein, A.: Hurwitz numbers and intersections on moduli spaces of curves. Invent. Math. 146(2), 297–327 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Goulden, I.P., Guay-Paquet, Mathieu, Novak, Jonathan: Toda equations and piecewise polynomiality for mixed double Hurwitz numbers. SIGMA Symmetr. Integr. Geom. Methods Appl. 12(paper 040), 10 (2016)Google Scholar
  17. 17.
    Goulden, I.P., Guay-Paquet, M., Novak, J.: Monotone Hurwitz numbers and the HCIZ integral. In Annales mathématiques Blaise Pascal, vol. 21, pp. 71–89 (2014)Google Scholar
  18. 18.
    Goulden, I.P., Jackson, D.M., Vakil, R.: Towards the geometry of double Hurwitz numbers. Adv. Math. 198(1), 43–92 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Guay-Paquet, M., Harnad, J.: Generating functions for weighted Hurwitz numbers (2014). arXiv:1408.6766
  20. 20.
    Hahn, M.A., Kramer, R., Lewanski, D.: Wall-crossing formulae and strong piecewise polynomiality for mixed grothendieck dessins d’enfant, monotone, and double simple hurwitz numbers. Adv. Math. 336, 38–69 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Hurwitz, A.: Ueber Riemann’sche Flächen mit gegebenen Verzweigungspunkten. Math. Ann. 39, 1–61 (1891)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Johnson, P.: Double Hurwitz numbers via the infinite wedge. Trans. Am. Math. Soc. 367(9), 6415–6440 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Kramer, R., Lewanski, D., Shadrin, S.: Quasi-polynomiality of monotone orbifold Hurwitz numbers and Grothendieck’s dessins d’enfants (2016). arXiv:1610.08376
  24. 24.
    Kazarian, M., Zograf, P.: Virasoro constraints and topological recursion for grothendieck’s dessin counting. Lett. Math. Phys. 105(8), 1057–1084 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Norbury, P.: String and dilaton equations for counting lattice points in the moduli space of curves (2009). arXiv:0905.4141
  26. 26.
    Shadrin, S., Shapiro, M., Vainshtein, A.: Chamber behavior of double Hurwitz numbers in genus 0. Adv. Math. 217(1), 79–96 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Woods, K.: The unreasonable ubiquitousness of quasi-polynomials. Electron. J. Comb. Res. Pap 21(1), 1–44 (2014)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Japan KK, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Mathematisches InstitutUniversität TübingenTübingenGermany

Personalised recommendations