Matrix integrals and gluing of polygons

We are interested in the number $T(k,g)$ of ways to glue pairwise all the edges of a $2k$-gon so as to produce a surface of a given genus $g$. For example, there is only one way to obtain a torus by gluing opposite sides of a square, and so $T(2,1)=1$. Spectral distribution moments for some random matrices ensembles can be expressed in terms of numbers $T(k,g)$. For example, if $H$ is a square $N\times N$ Hermitian Gaussian matrix, then
$$ \mathbf E\, \operatorname{Tr}H^{2k}=N^{k+1}\sum_{g=0}^{[k/2]} T(k,g)\frac{1}{N^{2g}}. $$

We discuss the connection between random matrices and gluings of polygons. Also we discuss Harer's and Zagier's theorem about how to compute the number $T(k,g)$.