Can anyone suggest a simple and efficient way (preferably embodied in computer code) to compute the average height function for lozenge tilings of an $a,b,c,a,b,c$ semiregular hexagon? 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 $a=b=c=1$ the average height function alternates between 1 and 2 on the boundary and is 3/2 in the middle.
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 probabilities (for the uniform distribution on dimer configurations); each of these probabilities can be written as a ratio of determinants, so I do know one way to compute the average height functions I'm interested in.
But I'm hoping for something simpler, along the lines of the recurrence formulas that apply to domino tilings of Aztec diamonds. Perhaps something like this follows from the recent work of Borodin (or maybe from the earlier work of Kuperberg), or can be obtained from Kuo's graphical condensation method. (And have I mentioned Speyer's paper on the octahedron recurrence? Ah, I just did.)