On the Lipschitz Constant of the RSK Correspondence

TL;DR

This paper investigates how much the Young diagram associated with a permutation via the RSK correspondence can change when the permutation is modified by a small number of transpositions, providing bounds and constructions.

Contribution

It establishes upper bounds on the Lipschitz constant of the RSK correspondence under transpositions and constructs permutations that nearly achieve these bounds.

Findings

Upper bounds on the Lipschitz constant as a function of transpositions

Explicit permutation constructions for t=1 achieving the bound

Near-matching permutations for larger t

Abstract

We view the RSK correspondence as associating to each permutation $\pi \in S_n$ a Young diagram $\lambda=\lambda(\pi)$, i.e. a partition of $n$. Suppose now that $\pi$ is left-multiplied by $t$ transpositions, what is the largest number of cells in $\lambda$ 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.