Algebraic structures defined on \(m\)-Dyck paths

We introduce natural binary set-theoretical products on the set of all \(m\)-Dyck paths, which led us to define a non-symmetric algebraic operad \(\Dy^m\), described on the vector space spanned by \(m\)-Dyck paths. Our construction is closely related to the \(m\)-Tamari lattice, so the products defi...

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Bibliographic Details
Published in:arXiv.org 2015-10
Main Authors: Daniel López N, Louis-François Préville-Ratelle, Ronco, María
Format: Article
Language:English
Subjects:
Algebra
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Summary:We introduce natural binary set-theoretical products on the set of all \(m\)-Dyck paths, which led us to define a non-symmetric algebraic operad \(\Dy^m\), described on the vector space spanned by \(m\)-Dyck paths. Our construction is closely related to the \(m\)-Tamari lattice, so the products defining \(\Dy^m\) are given by intervals in this lattice. For \(m=1\), 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 \(m\)-Dyck paths in the last section. As an additional result, for any composition of \(m+1\geq 2\) with \(r+1\) parts, we get a functor from the category of \(\Dy ^m\) algebras into the category of \(\Dy ^r\) algebras.
ISSN:2331-8422