Rational Dyck tilings

TL;DR

This paper introduces rational Dyck tilings, decomposes them into simpler components, and establishes combinatorial correspondences, advancing understanding of their structure and related permutation models.

Contribution

It presents the concept of rational Dyck tilings and a decomposition method that links them to well-studied combinatorial models, providing new insights.

Findings

Decomposition of rational Dyck tilings into (1,1)-Dyck tilings.

Establishment of a correspondence with $b$-Stirling permutations.

Representation of rational Dyck tilings as tuples of simpler tilings.

Abstract

We introduce rational Dyck tilings, or $(a,b)$-Dyck tilings, and study them by the decomposition into $(1,1)$-Dyck tilings. This decomposition allows us to make use of combinatorial models for $(1,1)$-Dyck tilings such as the Hermite history and the Dyck tiling strip bijection. Together with $b$-Stirling permutations associated to the rational Dyck tilings, we obtain a correspondence between an $(a,b)$-Dyck tiling and a tuple of $ab$ $(1,1)$-Dyck tilings.