TITLE: Partial duality of hypermaps.

Abstract.
A combinatorial description of (hyper)maps on orientable surfaces consists 
of two permutations from a symmetric group which is a Coxeter group of type 
$A_N$. It turns out that hypermaps on non orientable surfaces can be describe 
in a similar way replacing the Coxeter group $A_N$ by a Coxeter group of 
type $B_N$. I generalize the concept of partial duality of ribbon graphs to 
(hyper)maps and describe it in terms of the permutations. If time permits, I 
outline a relation with  Grothendieck's dessins d'enfants.