We study two combinatorial miracles relating purely poset-theoretic objects with purely Coxeter-theoretic objects. The first miracle is that there are the same number of linear extensions of the root poset as reduced words of the longest element (in certain types), while the second is that there are the same number of order ideals in the root poset as certain group elements (in more types). We place these miracles on remarkably similar footing in Theorem 3.1.9 and Conjecture 4.4.9.
Strangely, the first miracle is less general than the second. We motivate it in Chapter 3 by showing that behind each of its instances lurks a trivial result. This philosophy gives us a unified framework that collects several known theorems while allowing us to obtain new results.
We set up the language for the second miracle in Chapter 4, in which we review the many equivalent ways to define W-Catalan objects. Of note, we show how to interpret both the Kreweras complement and Cambrian rotation as certain walks on N. Reading's Cambrian lattice, which may be conjecturally mirrored on the nonnesting partitions.
In Chapter 5, we test the limits of the second miracle by turning inward to parabolic quotients. In doing so, we introduce a new generalization of W-Catalan combinatorics.