Brick polytopes, lattice quotients, and Hopf algebras

摘要: This paper is motivated by the interplay between Tamari lattice, J.-L. Loday's realization of associahedron, and Loday M. Ronco's Hopf algebra on binary trees. We show that these constructions extend in world acyclic $k$-triangulations, which were already considered as vertices V. Pilaud F. Santos' brick polytopes. describe combinatorially a natural surjection from permutations to $k$-triangulations. fibers this are classes congruence $\equiv^k$ $\mathfrak{S}_n$ defined transitive closure rewriting rule $U ac V_1 b_1 \cdots V_k b_k W \equiv^k U ca W$ for letters $a < b_1, \dots, c$ words $U, V_1, V_k, $[n]$. then increasing flip order $k$-triangulations lattice quotient weak congruence. Moreover, we use define subalgebra C. Malvenuto Reutenauer's permutations, indexed product coproduct its dual term combinatorial operations Finally, our results three directions, describing Cambrian, tuple, Schr\"oder version constructions.