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Séminaire Lotharingien de Combinatoire, B86a (2022), 15 pp.

# Tucker J. Ervin, Blake Jackson, Jay Lane, Kyungyong Lee, Son Dang Nguyen, Jack O'Donohue
and Michael Vaughan

# Permutations whose Reverse Shares the Same Recording Tableau in the RS Correspondence

**Abstract.**
The RS 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
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.

Received: September 30, 2021.
Revised: April 8, 2022.
Accepted: April 28, 2022.
Final Version: May 24, 2022.

The following versions are available: