Jones--Wenzl projections of type $D$ and Dyck tilings

We study the relation between a coefficient of an element of the Jones–Wenzl projection in the Temperley–Lieb algebra of type $D$ and an enumeration of Dyck tilings. The coefficient can be non-recursively expressed as an enumerative generating function of Dyck tilings by considering the generalized Hermite histories, which we call bi-colored vertical Hermite histories, on the tilings.