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A simplicial complex splitting associativity. (English) Zbl 1437.18014

A simplicial object of nonsymmetric operads \((\mathrm{Dyck}^m)_{m\geqslant 0}\) is introduced by generators and relations: \(\mathrm{Dyck}^m\) is generated by \(m+1\) binary products, with \(\dfrac{(m+1)(m+2)}{2}\) relations. in particular, \(\mathrm{Dyck}^0\) is the operad of associative algebras, and \(\mathrm{Dyck}^1\) is the operad of dendriform algebras. It is proved that the dimensions of the components of \(\mathrm{Dyck}^m\) are given by Fuss-Catalan numbers.
A notion of dendriform posets is introduced and it is proved that, firstly, such a family of posets gives a dendriform algebras and, secondly that \(m\)-simplexes in these posets give a \(\mathrm{Dyck}^m\)-algebras. Examples are given on surjections, permutations, planar binary trees or planar rooted trees, giving again well-known dendriform algebras.
Finally, the free \(\mathrm{Dyck}^m\)-algebra on one generator is combinatorially described in terms of \(m\)-Dyck paths, and it is proved that the \(m+1\) products in this algebraic structures are given by intervals in the \(m\)-Tamari lattices, generalizing Loday and Ronco’s result for the free dendriform algebra on planar binary trees.

MSC:

18M65 Non-symmetric operads, multicategories, generalized multicategories
06A11 Algebraic aspects of posets
16T30 Connections of Hopf algebras with combinatorics
05E45 Combinatorial aspects of simplicial complexes
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