Algebraic structures defined on -Dyck paths
arXiv:1508.01252
Abstract
We introduce natural binary set-theoretical products on the set of all -Dyck paths, which led us to define a non-symmetric algebraic operad $\Dy^m$, described on the vector space spanned by -Dyck paths. Our construction is closely related to the -Tamari lattice, so the products defining $\Dy^m$ are given by intervals in this lattice. For , we recover the notion of dendriform algebra introduced by J.-L. Loday in \cite{Lod}, and there exists a natural operad morphism from the operad ${\mbox {\it Ass}}$ of associative algebras into the operad $\Dy^m$, consequently $\Dy ^m$ is a Hopf operad. We give a description of the coproduct in terms of -Dyck paths in the last section. As an additional result, for any composition of with parts, we get a functor from the category of $\Dy ^m$ algebras into the category of $\Dy ^r$ algebras.
40 pages