2015/08/06 by N., Daniel López, Préville-Ratelle, Louis-François, Ronco, María
#05E05 #16T30 #Combinatorics (math.CO) #FOS: Mathematics #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.1508.01252
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 \Dym, described on the vector space spanned by m-Dyck paths. Our construction is closely related to the m-Tamari lattice, so the products defining \Dym are given by intervals in this lattice. For m=1, we recover the notion of dendriform algebra introduced by J.-L. Loday in \citeLod, and there exists a natural operad morphism from the operad \mbox \it Ass of associative algebras into the operad \Dym, 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≥ 2 with r+1 parts, we get a functor from the category of \Dy m algebras into the category of \Dy r algebras.