The algebra of binary search trees

Authors:
F. Hivert;J.-C. Novelli;J.-Y. Thibon
Affiliations:
Laboratoire Franco-Russe de Mathématiques, Independent University of Moscow, 11, Bolchoi Vlassesky per., 121002, Moscow, Russian Federation;Institut Gaspard Monge, Université de Marne-la-Vallée, 77454 Marne-la-Vallée cedex, France;Institut Gaspard Monge, Université de Marne-la-Vallée, 77454 Marne-la-Vallée cedex, France
Venue:
Theoretical Computer Science - Combinatorics on words
Year:
2005

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Abstract

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.