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The algebra of binary search trees. (English) Zbl 1072.05052
Summary: We introduce a monoid structure on the set of binary search trees, by a process very similar to the construction of the plactic monoid, the Robinson-Schensted insertion being replaced by the binary search tree insertion. This leads to a new construction of the algebra of planar binary trees of Loday-Ronco, defining it in the same way as non-commutative symmetric functions and free symmetric functions. We briefly explain how the main known properties of the Loday-Ronco algebra can be described and proved with this combinatorial point of view, and then discuss it from a representation theoretical point of view, which in turns leads to new combinatorial properties of binary trees.

MSC:
05E05 Symmetric functions and generalizations
68P05 Data structures
68R10 Graph theory (including graph drawing) in computer science
Software:
OEIS
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