Séminaitre équipe ADA du 4 février
The associative symmetric operad is an operad on permutations. It is an algebraic structure endowed with a composition operation allowing us to insert a permutation into another one. This structure is rich both under a combinatorial and an algebraic point of view. In this context, Aguiar and Livernet have constructed alternative bases of this operad relating it with the combinatorics of the weak order on permutations. In this talk, I will present a family of analogous operads, defined on some families of words of integers. These sets of words, called cliffs, can be put in correspondence with some usual combinatorial sets (permutations, increasing trees, Fuss-Catalan objects, etc.). The construction of this family of operads is detailed and some properties are presented. One of the peculiarities of some operads of this hierarchy is that, despite to their relative simplicity, some are infinitely generated and have nonquadratic and nonhomogeneous nontrivial relations.