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?