Brick polytopes, lattice quotients, and Hopf algebras

This paper is motivated by the interplay between the Tamari lattice, J.-L. Loday's realization of the associahedron, and J.-L. Loday and M. Ronco's Hopf algebra on binary trees. We show that these constructions extend in the world of acyclic k-triangulations, which were already considered as the vertices of V. Pilaud and F. Santos' brick polytopes. We describe combinatorially a natural surjection from the permutations to the acyclic k-triangulations. We show that the fibers of this surjection are the classes of the congruence ≡k on Sn defined as the transitive closure of the rewriting rule UacV1b1⋯VkbkW≡kUcaV1b1⋯VkbkW for letters a