A simplicial complex spliting associativity

Abstract: We introduce a simplicial object $({ \Dy^m}_{m\geq 0}, {\mathbb F}i, {\mathbb S}_j)$ in the category of non-symmetric algebraic operads, satisfying that $\Dy^0$ is the operad of associative algebras and $\Dy^1$ is J.-L. Loday\rq s operad of dendriform algebras. The dimensions of the operad $\Dy^m$ are given by the Fuss-Catalan numbers. Given a family of partially ordered sets ${\bold P}={P_n}{n\geq 1}$ we show that, under certain conditions, the vector space spanned by the set of $m$-simpleces of ${\bold P}$ is a $\Dy^m$ algebra. This construction, applied to certain combinatorial Hopf algebras, whose associative product comes from a dendriform structure, provides examples of $\Dy^m$ algebras.