Geodesic
Quasi-Polynomiality of Monotone Orbifold Hurwitz Numbers and Grothendieck's Dessins d'Enfants
Documenta mathematica, Tome 24 (2019), pp. 857-898.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

We prove quasi-polynomiality for monotone and strictly monotone orbifold Hurwitz numbers. The second enumerative problem is also known as enumeration of a special kind of Grothendieck's dessins d'enfants or $r$-hypermaps. These statements answer positively two conjectures proposed by Do-Karev and Do-Manescu. We also apply the same method to the usual orbifold Hurwitz numbers and obtain a new proof of the quasi-polynomiality in this case. In the second part of the paper we show that the property of quasi-polynomiality is equivalent in all these three cases to the property that the $n$-point generating function has a natural representation on the $n$-th cartesian powers of a certain algebraic curve. These representations are necessary conditions for the Chekhov-Eynard-Orantin topological recursion.
Classification : 14N10, 14H57, 05E05
Keywords: Hurwitz numbers, dessins d'enfants, spectral curves, enumerative geometry 
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     author = {Kramer, Reinier and Lewa\'nski, Danilo and Shadrin, Sergey},
     title = {Quasi-Polynomiality of {Monotone} {Orbifold} {Hurwitz} {Numbers} and {Grothendieck's} {Dessins} {d'Enfants}},
     journal = {Documenta mathematica},
     pages = {857--898},
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Kramer, Reinier; Lewański, Danilo; Shadrin, Sergey. Quasi-Polynomiality of Monotone Orbifold Hurwitz Numbers and Grothendieck's Dessins d'Enfants. Documenta mathematica, Tome 24 (2019), pp. 857-898. http://geodesic.mathdoc.fr/item/DOCMA_2019__24__a40/