Geodesic
Simplification of Formulas for the Number of Maps on Surfaces
Matematičeskie zametki, Tome 83 (2008) no. 1, pp. 14-23.

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Two combinatorial identities obtained by the author are used to simplify formulas for the number of general rooted cubic planar maps, for the number of $g$-essential maps on surfaces of small genus, and also for rooted Eulerian maps on the projective plane. Besides, an asymptotics for the number of maps with a large number of vertices is obtained.
Keywords: cubic planar map, $g$-essential map, surface of small genus, projective plane, Klein bottle, rooted Eulerian map, Euler beta function, gamma function, Stirling's formula. 
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V. A. Voblyi. Simplification of Formulas for the Number of Maps on Surfaces. Matematičeskie zametki, Tome 83 (2008) no. 1, pp. 14-23. http://geodesic.mathdoc.fr/item/MZM_2008_83_1_a1/

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