Geodesic
Order structure on the algebra of permutations and of planar binary trees
Journal of Algebraic Combinatorics, Tome 15 (2002) no. 3, pp. 253-270.

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Summary: Let $X _{n}$ be either the symmetric group on $n$ letters, the set of planar binary $n$-trees or the set of vertices of the $( n - 1)$-dimensional cube. In each case there exists a graded associative product on $oplus_{ $nge$0}$ K[ X $\_{n}$]. We prove that it can be described explicitly by using the weak Bruhat order on S $\_{n}$, the left-to-right order on planar trees, the lexicographic order in the cube case.
Keywords: planar binary tree, order structure, weak Bruhat order, algebra of permutations, dendriform algebra 
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     title = {Order structure on the algebra of permutations and of planar binary trees},
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Loday, Jean-Louis; Ronco, María O. Order structure on the algebra of permutations and of planar binary trees. Journal of Algebraic Combinatorics, Tome 15 (2002) no. 3, pp. 253-270. http://geodesic.mathdoc.fr/item/JAC_2002__15_3_a2/