Weak Bruhat order on the set of faces of the permutohedron and the associahedron

We define a partial order on the set of faces of the permutohedron, which extends the weak Bruhat order of the symmetric group. The restriction of this order gives a partially ordered set structure to the set of faces of the associahedron, which extends the Tamari order defined on the set of planar binary rooted trees. These orders are used to describe associative algebra structures on the vector spaces spanned by the set of faces of the permutohedra on one hand, and on the vector space spanned by the set of planar rooted trees on the other hand. Our results extend the description of the dendriform algebra structure on the spaces spanned by permutations and by planar binary trees in terms of the weak Bruhat order and the Tamari order, respectively, obtained by J.-L. Loday and the second author in [J.-L. Loday, M. Ronco, Order structure and the algebra of permutations and of planar binary trees, J. Algebraic Combin. 15 (2002) 253–270].