# Tableau Stabilization and Rectangular Tableaux Fixed by Promotion Powers

## TL;DR

This paper introduces tableau stabilization, a new concept based on jeu de taquin, and uses it to construct large rectangular tableaux fixed by promotion powers, connecting to cyclic sieving phenomena.

## Contribution

It develops the theory of tableau stabilization and applies it to explicitly construct fixed points of promotion powers in rectangular tableaux.

## Key findings

- Defined tableau stabilization and analyzed its properties.
- Established bounds and shape characteristics of stabilized tableaux.
- Constructed large rectangular tableaux fixed by promotion powers.

## Abstract

We introduce tableau stabilization, a new phenomenon and statistic on Young tableaux based on jeu de taquin. We investigate bounds for tableau stabilization, the shape of stabilized tableaux, and tableau stabilization as a permutation statistic. We apply tableau stabilization to construct the sufficiently large rectangular tableaux fixed by powers of promotion, which were counted by Brendon Rhoades via the cyclic sieving phenomenon