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Apr 1, 2012 at 16:03 comment added Benjamin Young $a \times b \times c$ boxed plane partitions can be described as families of $c$ nonintersecting lattice paths with $a+b$ steps. So a natural guess is to do your random point selection $c$ times, conditioned on the event that the lattice paths you get form a nonintersecting lattice path ensemble in the proper way. Does that work?
Mar 5, 2012 at 22:21 history asked James Propp CC BY-SA 3.0