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In other words, given a matrix $(p_{ij})$, is there a distribution on partitions of $[n]$ with the above property, and if so is there a way to sample from such distribution?
No; $ p_{1,2}=p_{1,3}=1$ implies
$p_{2,3}=1$.
No; $ p_{1,2}=p_{1,3}=1$ implies
$p_{2,3}=1$.
In other words, given a matrix $(p_{ij})$, is there a distribution on partitions of $[n]$ with the above property, and if so is there a way to sample from such distribution?
