Geodesic
Monotone orbifold Hurwitz numbers
Zapiski Nauchnykh Seminarov POMI, Combinatorics and graph theory. Part V, Tome 446 (2016), pp. 40-69

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In general, Hurwitz numbers count branched covers of the Riemann sphere with prescribed ramification data, or equivalently, factorisations in the symmetric group with prescribed cycle structure data. In this paper, we initiate the study of monotone orbifold Hurwitz numbers. These are simultaneously variations of the orbifold case and generalisations of the monotone case, both of which have been previously studied in the literature. We derive a cut-and-join recursion for monotone orbifold Hurwitz numbers, determine a quantum curve governing their wave function, and state an explicit conjecture relating them to topological recursion. 
@article{ZNSL_2016_446_a3,
     author = {N. Do and M. Karev},
     title = {Monotone orbifold {Hurwitz} numbers},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {40--69},
     publisher = {mathdoc},
     volume = {446},
     year = {2016},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2016_446_a3/}
}
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N. Do; M. Karev. Monotone orbifold Hurwitz numbers. Zapiski Nauchnykh Seminarov POMI, Combinatorics and graph theory. Part V, Tome 446 (2016), pp. 40-69. http://geodesic.mathdoc.fr/item/ZNSL_2016_446_a3/