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 
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/
@article{ZNSL_2013_417_a4,
     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},
     year = {2013},
     volume = {417},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2013_417_a4/}
}
TY  - JOUR
AU  - A. V. Pastor
TI  - On a gluing of surfaces of genus $g$ from 2 and 3 polygons
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2013
SP  - 128
EP  - 148
VL  - 417
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2013_417_a4/
LA  - ru
ID  - ZNSL_2013_417_a4
ER  - 
%0 Journal Article
%A A. V. Pastor
%T On a gluing of surfaces of genus $g$ from 2 and 3 polygons
%J Zapiski Nauchnykh Seminarov POMI
%D 2013
%P 128-148
%V 417
%U http://geodesic.mathdoc.fr/item/ZNSL_2013_417_a4/
%G ru
%F ZNSL_2013_417_a4

Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

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.

[1] N. M. Adrianov, “Analog formuly Kharera–Tsagira dlya odnokletochnykh dvukrashennykh kart”, Funkts. analiz i ego prilozh., 31:3 (1997), 1–9 | DOI | MR | Zbl

[2] N. V. Alexeev, J. E. Andersen, R. C. Penner, P. Zograf, Enumeration of chord diagrams on many intervals and their non-orientable analogs, 2013, arXiv: 1307.0967

[3] J. E. Andersen, R. C. Penner, C. M. Reidys, M. S. Waterman, Enumeration of linear chord diagrams, 2010, arXiv: 1010.5614

[4] J. E. Andersen, R. C. Penner, C. M. Reidys, R. R. Wang, Linear chord diagrams on two intervals, 2010, arXiv: 1010.5857

[5] R. Cori, A. Machì, “Maps, hypermaps and their automorphisms: a survey. I”, Exposition. Math., 10:5 (1992), 403–427 | MR | Zbl

[6] I. P. Goulden, A. Nica, “A direct bijection for the Harer–Zagier formula”, J. Combin. Theory Ser. A, 111:2 (2005), 224–238 | DOI | MR | Zbl

[7] I. P. Goulden, W. Slofstra, “Annular embeddings of permutations for arbitrary genus”, J. Combin. Theory Ser. A, 117 (2010), 272–288 | DOI | MR | Zbl

[8] V. A. Gurvich, G. B. Shabat, “Reshenie uravneniya Kharera–Tsagira”, Uspekhi mat. nauk, 48:1 (1993), 159–160 | MR | Zbl

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

[10] A. V. Pastor, O. P. Rodionova, “Nekotorye formuly dlya chisla skleek”, Zapiski nauchn. semin. POMI, 406, 2012, 117–156 | MR