Description
A large range of mathematical objects are enumerated by the famous $\text{Catalan}$ numbers : triangulation of polygons, non-crossing partitions, number of ways to correctly parenthesize an expression, $\text{Dyck}$ paths, binary trees, monotonic lattice paths, ... I will introduce a less famous object that is also enumerated by the $\text{Catalan}$ numbers : the $\text{Coxeter}$ sortable elements of the symmetric group. They are specific permutations that are defined $\textit{a priori}$ in a way completely unrelated to the $\text{Catalan}$ combinatorics. I will show explicit bijections from the $\text{Coxeter}$ sortable elements to the non-crossing partitions and to binary search trees and motivate their importance. Then, if time is permitted, I will present a generalization of the binary trees that emerges from the choice of specific $\text{Coxeter}$ elements and discuss.