On gluing a surface of genus $g$ from two and three polygons

TL;DR

This paper investigates the enumeration of ways to glue polygons into surfaces of genus g, providing elementary proofs and explicit formulas for cases involving two and three polygons, including the torus.

Contribution

It introduces elementary proofs and explicit formulas for counting gluings of polygons into surfaces of genus g, extending previous results to three polygons.

Findings

Elementary proof for the generating function of gluings from two polygons

Explicit formula for gluings from three polygons

Direct formula for gluings of a torus from three polygons

Abstract

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 [4]). 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.