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 tilings of hexagons, domino tilings of Aztec diamonds, and alternating-sign matrices. In each case, imposing appropriate boundary conditions on a finite region (Korepin calls them "domain-wall boundary conditions" while I might call them "steep boundary conditions") gives us a counting problem with a nice answer: $\prod\_{i=1}^{a} \prod\_{j=1}^b \prod\_{k=1}^c \frac{i+j+k-1}{i+j+k-2}$ (MacMahon), $2^{n(n+1)/2}$ (Elkies, Kuperberg, Larsen, and Propp), and $\prod\_{k=0}^{n-1} \frac{(3k+1)!}{(n+k)!}$ (Zeilberger found the first proof, Kuperberg found the first short proof). 


Question: Are there other members of this loosely-knit "family"? I once tried to find something analogous for Kagome ice, but failed. The twenty-vertex model (orientations of a triangular lattice so that each vertex has three arrows pointing in and three pointing out) also didn't lead to anything. But just because the obvious sorts of steep boundary conditions didn't give rise to enumerations I recognized as "nice" doesn't mean that there aren't theorems out there.


I should say that, although the nicest answers are of the sort given by the three expressions above (with numbers that have only "small" prime factors), even an answer like a triple sum of binomial coefficients is more than we could reasonably expect Nature to hand us, and so should be considered "nice". (I know an example like this involving lozenge tilings of octagons, due to Destainville, Mosseri, and Bailly.)


Since I mentioned stat mech in the title, I should stress two ways in which my topic is different from what physicists mean by "exactly solvable models": (1) Physicists are interested in the asymptotic growth-rate of the number of configurations (the entropy), and usually not interested in exact combinatorial formulas for their own sake. (2) Domain-wall boundary conditions aren't the boundary conditions physicists care about most.



Exact enumerations from two-dimensional stat mech models
Jim already knows this very well but I'll state for the record that there are a number of nontrivial variants of the EKLP result, involving dimer placements on other two dimensional regular grids. Ciucu's [Perfect matchings and perfect powers](http://arxiv.org/abs/math/0501521) is a good survey and I added a few more to the list in [Perfect matchings and the octahedron recurrence](http://arxiv.org/abs/math/0402452). I talked with Rick Kenyon about these examples, and we agreed that they basically seem to show up when the spectral curve (as in [Kenyon-Okounkov-Sheffield](http://arxiv.org/abs/math-ph/0311005)) has genus zero. I'm not prepared to state a precise result here, though. (Perhaps Rick is?)


I view all these dimer results as much closer to each other, and much easier, than the MacMahon or Zeilberger/Kuperberg results. It sounds like Jim is looking for things outside the dimer realm, so this won't satisfy him, but I figured I'd leave this note in case someone else stumbles upon this question in their search results.


