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

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