On a gluing of surfaces of genus $g$ from 2 and 3 polygons

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.