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I want to gather facts and questions related to 3D generalizations of permutations, RSK correspondence, contingency tables, etc. One reason I am interested in this is because it is potentially related ...

Let $(X_1,X_2,\ldots)$ be a sequence of i.i.d. random variables. It is known that if these random variables are distributed uniformly on the unit interval, then applying the RSK algorithm to this ...