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Let iXn be either the symmetric group on in letters, the set of planar binary in-trees or the set of vertices of the (in − 1)-dimensional cube. In each case there exists a graded associative product on ⊕in≥0iK[iXn]. We prove that it can be described explicitly by using the weak Bruhat order on iSn, the left-to-right order on planar trees, the lexicographic order in the cube case.

Order Structure on the Algebra of Permutations and of Planar Binary Trees