Dyck tilings of type $D$

TL;DR

This paper introduces and analyzes Dyck tilings of type D, connecting their generating functions to ballot tilings of type B, and develops combinatorial structures like link patterns and plane trees to compute these functions.

Contribution

It presents the first study of cover-inclusive and cover-exclusive Dyck tilings of type D, linking their generating functions to those of ballot tilings of type B, and constructs new combinatorial maps.

Findings

Generating functions of type D Dyck tilings expressed via type B ballot tilings.

Introduction of link patterns and plane trees for type D Dyck tilings.

Construction of a map from trees to $ ext{Z}[q]$ for generating functions.

Abstract

We introduce and study cover-inclusive and cover-exclusive Dyck tilings of type $D$. It is shown that the generating functions of Dyck tilings of type $D$ are expressed in terms of the generating function of ballot tilings of type $B$. We introduce link patterns of type $D$ and plane trees for a ballot path, and construct a map from trees to $\mathbb{Z}[q]$. This map gives the generating function of cover-inclusive Dyck tilings of type $D$ associated to the ballot path.