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Séminaire Lotharingien de Combinatoire, 84B.22 (2020), 12 pp.

# Camille Combe and Samuele Giraudo

# Three Interacting Families of Fuss-Catalan Posets

**Abstract.**
Three families of posets depending on a nonnegative integer parameter *m* are
introduced. The underlying sets of these posets are enumerated by the *m*-Fuss-Catalan
numbers. Among these, one is a generalization of Stanley lattices and another one is a
generalization of Tamari lattices. The three families of posets are related: they fit into a
chain for the order extension relation and they share some properties. Two associative
algebras are constructed as quotients of generalizations of the Malvenuto-Reutenauer
algebra. Their products describe intervals of our analogues of Stanley lattices and Tamari
lattices. In particular, one is a generalization of the Loday-Ronco algebra.

Received: November 20, 2019.
Accepted: February 20, 2020.
Final version: April 30, 2020.

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