Geodesic
Gluings of Surfaces with Polygonal Boundaries
Funkcionalʹnyj analiz i ego priloženiâ, Tome 43 (2009) no. 4, pp. 3-13

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By pairwise gluing edges of a polygon, one obtains two-dimensional surfaces with handles and holes. We compute the number $\mathcal{N}_{g,L}(n_1,\dots,n_L)$ of distinct ways to obtain a surface of given genus $g$ whose boundary consists of $L$ polygonal components with given numbers $n_1,\dots,n_L$ of edges. Using combinatorial relations between graphs on real two-dimensional surfaces, we derive recursion relations between the $\mathcal{N}_{g,L}$. We show that the Harer–Zagier numbers arise as a special case of $\mathcal{N}_{g,L}$ and derive a new closed-form expression for them.
Keywords: graph on surface, number of graphs, generating function. 
@article{FAA_2009_43_4_a1,
     author = {E. T. Akhmedov and Sh. R. Shakirov},
     title = {Gluings of {Surfaces} with {Polygonal} {Boundaries}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {3--13},
     publisher = {mathdoc},
     volume = {43},
     number = {4},
     year = {2009},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2009_43_4_a1/}
}
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E. T. Akhmedov; Sh. R. Shakirov. Gluings of Surfaces with Polygonal Boundaries. Funkcionalʹnyj analiz i ego priloženiâ, Tome 43 (2009) no. 4, pp. 3-13. http://geodesic.mathdoc.fr/item/FAA_2009_43_4_a1/