Vol. 14, No. 3, 2021

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Iterating the RSK bijection

Maria Gillespie, Jacob Hocevar, Ananya Kulshrestha and Kosha Upadhyay

Vol. 14 (2021), No. 3, 475–494

We investigate the dynamics of the well-known RSK bijection on permutations when iterated on various reading words of the recording tableau. In the setting of the ordinary (row) reading word, we show that there is exactly one fixed point per partition shape, and that it is always reached within two steps from any starting permutation.

We also consider the modified dynamical systems formed by iterating RSK on the column reading word and the reversed reading word of the recording tableau. We show that the column reading word gives similar dynamics to the row reading word. On the other hand, for the reversed reading word, we always reach either a 2-cycle or fixed point after two steps. In fact, we reach a fixed point if and only if the shape of the initial tableau is self-conjugate.

Young tableaux, permutations, dynamical systems, RSK correspondence
Mathematical Subject Classification
Primary: 05A05
Secondary: 05E99
Received: 1 October 2020
Revised: 11 February 2021
Accepted: 26 February 2021
Published: 17 July 2021

Communicated by Kenneth S. Berenhaut
Maria Gillespie
Department of Mathematics
Colorado State University
Fort Collins, CO
United States
Jacob Hocevar
McKinney Christian Academy
Plano, TX
United States
Ananya Kulshrestha
Irvington High School
Fremont, CA
United States
Kosha Upadhyay
Bellevue High School
Bellevue, WA
United States