Abstract
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 ⊕ n≥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.
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Loday, JL., Ronco, M.O. Order Structure on the Algebra of Permutations and of Planar Binary Trees. Journal of Algebraic Combinatorics 15, 253–270 (2002). https://doi.org/10.1023/A:1015064508594
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DOI: https://doi.org/10.1023/A:1015064508594