http://arxiv.org/abs/0712.2448

Gluing of Surfaces with Polygonal Boundaries E. T. Akhmedov, Sh. Shakirov

By pairwise gluing of edges of a polygon, o produces two-dimensional surfaces with handles and boundaries. In this paper, we count the number ${\cal N}_{g,L}(n_1, n_2, n_L)$ of different ways to produce a surfac of given genus $g$ with $L$ polygonal boundaries with given numbers of edges $n_1, n_2, >..., n_L$. Using combinatorial relations between graphs on real two-dimensional surfaces, we derive recursive relations between ${\cal N}_{g,L}$. We show that Harer-Zagier numbers appear as a particular case of ${\cal N}_{g,L}$ and deriv a new explicit expression for them. Comments: 7 pages, 9 figures

It seems proposes quite elementary proof. The key idea that they found some generalization which is more easy to prove.

http://arxiv.org/abs/0712.2448

Gluing of Surfaces with Polygonal Boundaries E. T. Akhmedov, Sh. Shakirov

By pairwise gluing of edges of a polygon, o produces two-dimensional surfaces with handles and boundaries. In this paper, we count the number ${\cal N}_{g,L}(n_1, n_2, n_L)$ of different ways to produce a surfac of given genus $g$ with $L$ polygonal boundaries with given numbers of edges $n_1, n_2, >..., n_L$. Using combinatorial relations between graphs on real two-dimensional surfaces, we derive recursive relations between $\cal N_{g,L}$. We show that Harer-Zagier numbers appear as a particular case of ${\cal N}_{g,L}$ and deriv a new explicit expression for them. Comments: 7 pages, 9 figures

It seems proposes quite elementary proof. The key idea that they found some generalization which is more easy to prove.