The RS(K) correspondence is a bijection between elements of the symmetric group $S_n$ and pairs of standard tableaux of the same shape. The symmetric group is partially ordered by the Bruhat order, so this bijection induces a partial ordering "$\leq$"on the set of pairs of standard tableaux. Is a natural "tableaux-theoretic" description of this ordering on pairs of tableaux known? That is, can we give a condition for $(S,T)\leq (S',T')$ more natural than "apply RSK in reverse and see if the second permutation dominates the first in the Bruhat order"?
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