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

TL;DR

This paper explores the connection between Jones--Wenzl projections in type D Temperley--Lieb algebras and Dyck tilings, providing a non-recursive formula via bi-colored Hermite histories.

Contribution

It introduces bi-colored vertical Hermite histories to express Jones--Wenzl projection coefficients as generating functions of Dyck tilings, advancing combinatorial understanding.

Findings

Coefficient expressed as a generating function of Dyck tilings

Introduction of bi-colored Hermite histories

Non-recursive formula for Jones--Wenzl projections

Abstract

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.