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Aztec diamond of order $n$ or random lozenge tilings of the regular hexagon of order $n$, what's the typical order of magnitude of the area of the symmetric difference between the temperate zone of the tiling …

hinges on a combinatorial lemma relating the number of vertical dominos in a tiling to the parity of the permutation-matrix associated with the tiling. What's the most direct way to see this? … My favorite approach is to set up a distributive lattice structure on the set of tilings, and use it to show that every tiling may be obtained from every other by rotating 2-by-2 blocks, and then use this …

Exact enumerations corresponding to the dimer model on a hexagonal grid, the dimer model on a square grid, and the four-vertex (aka square ice) model on a square grid are known, namely: lozenge tiling …

I prefer to scale my height functions so that the height-change along each tile-edge is 1 and so that the lowest possible height of any vertex in any tiling of the region is 0, so that for instance when … When we replace the tiling model by the (dual) dimer model, height becomes associated with faces rather than vertices, and the average height of a face can be written as a linear combination of edge-inclusion …

Consider the "fortress graph" of order $n$ (see Figure 9 of http://faculty.uml.edu/jpropp/tiling/www/mdblum/arctic.html). …

On the one hand, it seems plausible that some sort of greedy approach to fitting together translates of $B$ might be guaranteed to yield a tiling if one exists (and to terminate quickly if none exists) … , much as is the case for two-dimensional tiling. …