6
$\begingroup$

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?

$\endgroup$
1
  • $\begingroup$ $a \times b \times c$ 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? $\endgroup$ Apr 1, 2012 at 16:03

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.