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?