Generalized Dyck tilings

TL;DR

This paper explores generalized Dyck tilings, connecting them with permutation orders and Stirling permutations, and provides enumeration results for these new classes of tilings.

Contribution

It introduces $k$-Dyck tilings and symmetric Dyck tilings, establishing their relation to permutation orders and providing enumeration methods.

Findings

Bijection between $k$-Dyck tilings and intervals in permutation orders

Enumeration formulas for symmetric Dyck tilings

New perspective on Dyck tilings via permutation theory

Abstract

Recently, Kenyon and Wilson introduced Dyck tilings, which are certain tilings of the region between two Dyck paths. The enumeration of Dyck tilings is related with hook formulas for forests and the combinatorics of Hermite polynomials. The first goal of this work is to give an alternative point of view on Dyck tilings by making use of the weak order and the Bruhat order on permutations. Then we introduce two natural generalizations: $k$-Dyck tilings and symmetric Dyck tilings. We are led to consider Stirling permutations, and define an analog of the Bruhat order on them. We show that certain families of $k$-Dyck tilings are in bijection with intervals in this order. We also enumerate symmetric Dyck tilings.