Weak Bruhat order on the set of faces of the permutahedra

We extend the weak Bruhat order of a finite Coxeter group to the set of its coclasses, modulo parabolic standard subgroups. We use this order to describe associative algebra structures on the vector spaces spanned by the faces of permutahedra on one side, and on the vector space spanned by the planar rooted trees, on the other one.