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. 
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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/

[1] J. Harer, D. Zagier, “The Euler characteristic of the moduli space of curves”, Invent Math., 85:3 (1986), 457–485 | DOI | MR | Zbl

[2] S. K. Lando, Lektsii o proizvodyaschikh funktsiyakh, MTsNMO, 2002

[3] S. Lando, A. Zvonkin, Graphs on Surfaces and Their Applications, Encyclopedia of Math. Sciences, 141, Springer-Verlag, 2004 | MR | Zbl

[4] J. Stasheff, “Homotopy associativity of H-spaces I”, Trans. Amer. Math. Soc., 108 (1963), 275–292 | DOI | MR | Zbl

[5] I. Gelfand, M. Kapranov, A. Zelevinsky, Discriminants, Resultants and Multidimensional Determinants, Birkhauser Boston, Boston, MA, 1994 | MR | Zbl

[6] M. Kontsevich, “Feynman diagrams and low-dimensional topology”, First European Congress of Math., Vol. II (Paris, 1992), Progr. Math., 120, Birkhauser, Basel, 1994, 97–121 | MR | Zbl

[7] M. L. Kontsevich, “Teoriya peresechenii na prostranstve modulei krivykh”, Funkts. analiz i ego pril., 25:2 (1991), 50–57 | MR | Zbl