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If one picks $a+b$ points uniformly at random (and independently) in $[0,1]$, coloring $a$ of them Red and $b$ of them Blue, then reading the points from left to right one gets a uniform random sequence of $a$ R's and $b$ B's. That is, we have a hyperplane arrangement on {$(x_1,x_2,\dots,x_{a+b})$} $\in [0,1]^{a+b}$ with planes given by the equations $x_i = x_j$ with $i \leq a < j$, the cells of which have equal measure and are in correspondence with the lattice paths from $(0,0)$ to $(a,b)$, which in turn are in correspondence with partitions whose Young diagram fits in an $a$-by-$b$ rectangle.

Can anything analogous (but of course more complicated) be done with 3-dimensional Young diagrams in an $a$-by-$b$-by-$c$ box?

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$a \times b \times c$ 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? – Benjamin Young Apr 1 '12 at 16:03

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