Geodesic
Recent progress in enumeration of hypermaps
Zapiski Nauchnykh Seminarov POMI, Combinatorics and graph theory. Part V, Tome 446 (2016), pp. 139-164
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We enumerate the isomorphism classes of hypermaps of a given genus $g\le6$ and a given number of darts $d$. The hypermaps of a given genus $g$ are distinguished up to orientation preserving isomorphisms. Our results depend on recent progress in counting rooted hypermaps, in particular by P. Zograf, M. Kazarian, A. Giorgetti and T. Walsh. These results can be interpreted as an enumeration of conjugacy classes of subgroups of the free Fuchsian group of rank two with a genus restriction. 
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A. Mednykh; R. Nedela. Recent progress in enumeration of hypermaps. Zapiski Nauchnykh Seminarov POMI, Combinatorics and graph theory. Part V, Tome 446 (2016), pp. 139-164. http://geodesic.mathdoc.fr/item/ZNSL_2016_446_a7/

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