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Three Fuss-Catalan posets in interaction and their associative algebras

We introduce $δ$-cliffs, a generalization of permutations and increasing trees depending on a range map $δ$. We define a first lattice structure on these objects and we establish general results about its subposets. Among them, we describe sufficient conditions to have EL-shellable posets, lattices with algorithms to compute the meet and the join of two elements, and lattices constructible by interval doubling. Some of these subposets admit natural geometric realizations. Then, we introduce three families of subposets which, for some maps $δ$, have underlying sets enumerated by the Fuss-Catalan numbers. Among these, one is a generalization of Stanley lattices and another one is a generalization of Tamari lattices. These three families of posets fit into a chain for the order extension relation and they share some properties. Finally, in the same way as the product of the Malvenuto-Reutenauer algebra forms intervals of the right weak Bruhat order of permutations, we construct algebras whose products form intervals of the lattices of $δ$-cliff. We provide necessary and sufficient conditions on $δ$ to have associative, finitely presented, or free algebras. We end this work by using the previous Fuss-Catalan posets to define quotients of our algebras of $δ$-cliffs. In particular, one is a generalization of the Loday-Ronco algebra.