  In "Hopf algebra of the planar binary trees", Adv. Math. 139 (1998), no. 2,
293--309, we constructed by induction a graded associative product on the
vector space generated by the planar binary trees (resp. the permutations). In
the present paper we prove an explicit formula for this product in terms of a
partial order on the set of planar binary trees (resp. the weak Bruhat order on
the symmetric groups).
