Rational Dyck tilings

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.