Mixed Dimer Models for Euler and Catalan Numbers

We study the enumeration of mixed dimer covers on skew Young diagrams of ribbon shape (also called border strips or snake graphs). For the two extreme cases of straight and zigzag shapes, we show that the number of mixed dimer covers are given by the Euler and Catalan numbers. We also give q-analogs by showing that the rank generating functions of the partial orders on mixed dimer covers agree with certain q-Euler and q-Catalan numbers. These q-analogs are a consequence of an isomorphism between the partial order on mixed dimer covers and the so-called middle order on certain classes of permutations.