Proving the number of permutations $A,B\;$ with $n+1$ total cycles and $AB=(123\cdots n)$ is $C_n$ Please give a combinatorial proof of the following:
The number of pairs $(A, B)$ of permutations of the set $\{ 1, 2,\ldots,n \}$ such that  they have a total of $n+1$ cycles and their composition $AB$ is the permutation $(123 \ldots n)$, is the $n$th catalan number.

Take a plane tree with $n+1$ vertices, take a bipartition and associate to each vertex the cyclic permutation that changes the adjacent edges cyclically. 
The products of the cycles of each bipartition give the two permutations.
This is a restriction of a bijection by Cori for planar maps.
