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<b1,…,bk<c and words U,V1,…,Vk,W on [n]. We then
show that the increasing flip order on k-triangulations is the lattice
quotient of the weak order by this congruence. Moreover, we use this surjection
to define a Hopf subalgebra of C. Malvenuto and C. Reutenauer's Hopf algebra on
permutations, indexed by acyclic k-triangulations, and to describe the
product and coproduct in this algebra and its dual in term of combinatorial
operations on acyclic k-triangulations. Finally, we extend our results in
three directions, describing a Cambrian, a tuple, and a Schr\"oder version of
these constructions.Comment: 59 pages, 32 figure