Jones--Wenzl projections and Dyck tilings: type $A$ and $B$

Abstract

The Jones--Wenzl projections are a special class of elements of the Temperley--Lieb algebra. We prove that the coefficient appearing in the Jones--Wenzl projection is given by a generating function of combinatorial objects, called Dyck tilings. We also show that this correspondence holds for type $B$ case.