Weak order, hyperplane arrangements, and the Tamari lattice

Abstract

One of the most elegant ways in which the Eulerian numbers and the Narayana numbers arise is in the counting of faces of polytopes. These polytopes are related to certain poset structures on, respectively, the set of all permutations of [n] and on the set of 231-avoiding permutations of [n] (or any set of Catalan objects). These posets are known as the weak order and the Tamari lattice, respectively. Our study of the weak order leads naturally to a side trip into the realm of hyperplane arrangements. This geometric perspective will be useful to have in later parts of the book.