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

TL;DR

This paper characterizes permutations whose reverse shares the same recording tableau in the RSK correspondence, proving an explicit enumeration formula based on the shape of the tableau.

Contribution

It establishes that such permutations correspond to recording tableaux with symmetric hook shapes and proves the exact count formula for these permutations.

Findings

Permutations with symmetric hook-shaped recording tableaux satisfy Q(w) = Q(w^r).

The number of such permutations is given by a specific combinatorial formula for odd n.

The shape symmetry condition is key to characterizing and counting these permutations.

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.