Geodesic
On a~gluing of surfaces of genus~$g$ from~2 and~3 polygons
Zapiski Nauchnykh Seminarov POMI, Combinatorics and graph theory. Part VI, Tome 417 (2013), pp. 128-148

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In this paper, the number of ways to glue together several polygons into a surface of genus $g$ has been investigated. We've given an elementary proof on the formula for the generating function $\mathbf C_g^{[2]}(z)$ of the number of gluings surface of genus $g$ from two polygons (see also R. C. Penner et al. {\it Linear chord diagrams on two intervals. (2010), arXiv:1010.5857). Moreover, we've proven a similar formula for gluings surface of genus $g$ from three polygons. As a corollary, we've proven a direct formula for the number of gluings torus from three polygons. 
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     author = {A. V. Pastor},
     title = {On a~gluing of surfaces of genus~$g$ from~2 and~3 polygons},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {128--148},
     publisher = {mathdoc},
     volume = {417},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2013_417_a4/}
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A. V. Pastor. On a~gluing of surfaces of genus~$g$ from~2 and~3 polygons. Zapiski Nauchnykh Seminarov POMI, Combinatorics and graph theory. Part VI, Tome 417 (2013), pp. 128-148. http://geodesic.mathdoc.fr/item/ZNSL_2013_417_a4/