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On the Lipschitz Constant of the RSK Correspondence

We view the RSK correspondence as associating to each permutation $π\in S_n$ a Young diagram $λ=λ(π)$, i.e. a partition of $n$. Suppose now that $π$ is left-multiplied by $t$ transpositions, what is the largest number of cells in $λ$ that can change as a result? It is natural refer to this question as the search for the Lipschitz constant of the RSK correspondence. We show upper bounds on this Lipschitz constant as a function of $t$. For $t=1$, we give a construction of permutations that achieve this bound exactly. For larger $t$ we construct permutations which come close to matching the upper bound that we prove.