Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia named after Patrice Lumumba (RUDN University)8339ArticlesСтатьиConference Report, Theses of ReportMatrix Integrals and Gluings of Regular 2n-gonsМатричные интегралы и склейки правильных 2n-угольниковShishaninA OШишанинАндрей ОлеговичКафедра физикиshishandr@rambler.ruState Technical University n.a. N.E. BaumanМосковский государственный технический университет им. Н.Э. Баумана150120131NO1 (2013)№1 (2013)28428708092016Copyright ©; 2013, Шишанин А.О.2013Шишанин А.О.http://creativecommons.org/licenses/by/4.0https://journals.rudn.ru/miph/article/view/8339This review is concerned applications of matrix models in combinatorics. We will discuss counting of orientable and nonorientable gluings of regular 2n-gons using gaussian matrix integrals.Рассматриваются приложения матричных моделей в комбинаторике. Обсуждается подсчёт ориентируемых и неориентируемых склеек правильных 2n-угольников с помощью гауссовых интегралов по ортогональным матрицамmatrix integralsgeneralized Catalan numbersgenerating function of gluingsvirtual Euler characteristicматричные интегралыобобщённые числа Каталанапроизводящая функция склееквиртуальная эйлерова характеристика1.’t Hooft G. A Planar Diagram Theory for Strong Interactions // Nucl. Phys. — 1974. — Vol. 9, No 6. — Pp. 461–473.2.Harer J., Zagier D. The Euler Characteristics of the Moduli Space of Curves // Invent. Math. — 1986. — Vol. 85. — Pp. 457–485.3.Zvonkin A.K., Lando S.K. Graphs on Surfaces and Their Applications. — Springer, 2003. — Vol. 459.4.Alexandrov A., Mironov A., Morozov A. Partition Functions of Matrix Models as the First Special Functions of String Theory I. Finite Size Hermitean 1-Matrix Model // Int.J.Mod.Phys. A. — 2004. — Vol. 19. — Pp. 4127–4165.5.Adrianov N.M. An Analog of the Harer–Zagier Formula for Unicellular Bicolored Maps // Funk. Analys and its Appl. — 1997. — Vol. 31. — Pp. 1–9.6.Akhmedov E.T., Shakirov S. Gluings of Surfaces with Polygonal Boundaries // Funk. Analys and its Appl. — 2009. — Vol. 43, issue 4. — Pp. 3–13.7.Morozov A.and Shakirov S. Exact 2-Point Function in Hermitian Matrix Model // J. of High Energy Phys. — 2009. — Vol. 0912. — Pp. 002–033.8.Lass B. Demonstration Combinatoire de la Formule de Harer–Zagier // C. R. Acad. Sci. Paris. — 2009. — Vol. 0912. — Pp. 002–033.9.Penner R.C. The Moduli Space of a Punctured Surface and Perturbative Series // Bull. Amer. Math. Soc. — 1986. — Vol. 15. — Pp. 73–77.10.Distler J., Vafa C. A Critical Matrix Model at. = 1 // Mod. Phys. Lett. A. — 1991. — Vol. 6. — Pp. 259–270.11.Kontsevich M. Intersection Theory on the Moduli Space of Curves and the Matrix Airy Function // Comm. Math. Phys. — 1992. — Vol. 147. — Pp. 1–23.12.Golden I., Jackson D. Maps in Locally Orientable Surfaces and Integrals over Real Symmetric Surfaces // Can. J. Math. — 1997. — Vol. 49. — Pp. 865–882.13.Harris G., Martinec E. Unoriented Strings and Matrix Ensembles // Phys. Lett. B. — 1990. — Vol. 245. — Pp. 384–392.14.Brezin E., Neuberger H. Multicritical Points of Unoriented Random Surfaces // Nucl. Phys. B. — 1991. — Vol. 350. — Pp. 513–553.15.Chekhov L., Zabrodin A. A Critical Matrix Model for Nonoriented String // Mod. Phys. Lett. — 1991. — Vol. 6. — Pp. 3143–3152.16.Ledoux M. A Recursion Formula for the Moments of the Gaussian Orthogonal Ensemble // Ann. Inst. H. Poincar.e Probab. Statist. — 2009. — Vol. 45. — Pp. 754–769.17.Bernardi O. An Analogue of the Harer–Zagier Formula for Unicellular Maps on General Surfaces // Adv. in Appl. Math. — 2012. — Vol. 48. — Pp. 164–180.