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arXiv · 2108.08657

Permutations whose reverse shares the same recording tableau in the RSK correspondence

Tucker J. Ervin·Blake Jackson·Jay Lane·Kyungyong Lee·Son Dang Nguyen·Jack O'Donohue·Michael Vaughan
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Abstract

The RSK correspondence is a bijection between permutations and pairs of standard Young tableaux with identical shape, where the tableaux are commonly denoted $P$ (insertion) and $Q$ (recording). It has been an open problem to demonstrate $$ |\{w \in \mathfrak{S}_n | \, Q(w) = Q(w^r)\}| = \begin{cases} \displaystyle 2^{\frac{n-1}{2}}{n-1 \choose \frac{n-1}{2}} & n \text{ odd} \newline \displaystyle 0 & n \text{ even} \end{cases}, $$ where $w^r$ is the reverse permutation of $w$. First we show that for each $w$ where $Q(w) = Q(w^r)$ the recording tableau $Q(w)$ has a symmetric hook shape and satisfies a certain simple property. From these two results, we succeed in proving the desired identity.

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Tucker J. Ervin, Blake Jackson, Jay Lane, Kyungyong Lee, Son Dang Nguyen, Jack O'Donohue, Michael Vaughan. 2021-08-19. Permutations whose reverse shares the same recording tableau in the RSK correspondence. https://arxiv.org/abs/2108.08657

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