Permutrees

International audienceWe introduce permutrees, a unified model for permutations, binary trees, Cambrian trees and binary sequences. In this extended abstract, we summarize our main results describing combinatorial and geometric properties of permutrees, in particular the permutree lattices (specializing to the weak order, the Tamari, the Cambrian and the boolean lattice) and permutreehedra (specializing to the permutahedron, the associahedra, and the cube)