In a 1986 paper, Harer and Zagier proved the recursion:


$$(n+1)e(g,n)=(4n-2)e(g,n-1)+(2n-1)(n-1)(2n-3)e(g-1,n-2)$$


where e(g,n) is the number of ways of grouping sides $S\_1...S\_{2n}$ of a 2n-gon into n pairs, such that, after identifying pairs with the corresponding orientation one gets an orientable surface of genus g.


Their proof is really indirect, it involves pages of calculations. 


Is there a simple proof of this fact?



A direct proof of the Harer-Zagier recursion enumerating the ways to paste a 2n-gon to get a genus g surface? 
<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.


