  We study the number of tilings of skew Young diagrams by ribbon tiles shaped
like Dyck paths, in which the tiles are "vertically decreasing". We use these
quantities to compute pairing probabilities in the double-dimer model: Given a
planar bipartite graph $G$ with special vertices, called nodes, on the outer
face, the double-dimer model is formed by the superposition of a uniformly
random dimer configuration (perfect matching) of $G$ together with a random
dimer configuration of the graph formed from $G$ by deleting the nodes. The
double-dimer configuration consists of loops, doubled edges, and chains that
start and end at the boundary nodes. We are interested in how the chains
connect the nodes. An interesting special case is when the graph is
$\epsilon(\Z\times\N)$ and the nodes are at evenly spaced locations on the
boundary $\R$ as the grid spacing $\epsilon\to 0$.
