Order types of reflection orders on affine Weyl groups

In this paper, we investigate the order types of reflection orders on irreducible affine Weyl groups. We show that they are intimately related to the Catalan combinatorics. We explicitly describe all of those order types and show that these order types can be parameterized by deformed Dyck words, which can be enumerated by Catalan numbers.
Comment: 21 pages; add some explanatory remarks and pictures to improve readability; add an appendix on the enumeration of deformed Dyck words