# Cliff posets and algebras

Samuele Giraudo (IGM, Paris-Est Marne la Vallée)The Malvenuto-Reutenauer Hopf bialgebra is a very central object in algebraic combinatorics since this structure admits as quotients, subalgebras, or generalizations a large hierarchy of related other structures. This algebra is defined on the linear span of all permutations and its combinatorial heart is based upon the so-called shifted shuffle operation of permutations. There are also many links between this structure and the right weak order, which is a lattice on permutations. In this work, we build an analogous algebra based on a variant of the right weak order. For this, we consider in fact new combinatorial objects inspired from Lehmer codes called cliffs. We obtain lattice structures, magmatic and/or associative algebras, and generalizations of Tamari lattices and of Stanley lattices, all parametrized by infinite words of nonnegative integers. We get by the way generalizations of the Loday-Ronco algebra of binary trees. In this talk, we will present these objects and some of their properties.

This is joint work with Camille Combe (PhD student at IRMA, Strasbourg).