13,655 research outputs found
The Algebra of Binary Search Trees
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.Comment: 49 page
Set-theoretic solutions of the Yang-Baxter equation, RC-calculus, and Garside germs
Building on a result by W. Rump, we show how to exploit the right-cyclic law
(x.y).(x.z) = (y.x).(y.z) in order to investigate the structure groups and
monoids attached with (involutive nondegenerate) set-theoretic solutions of the
Yang-Baxter equation. We develop a sort of right-cyclic calculus, and use it to
obtain short proofs for the existence both of the Garside structure and of the
I-structure of such groups. We describe finite quotients that exactly play for
the considered groups the role that Coxeter groups play for Artin-Tits groups
- …