Tableau stabilization and lattice paths

If one attaches shifted copies of a skew tableau to the right of itself and rectifies, at a certain point the copies no longer experience vertical slides, a phenomenon called tableau stabilization. While tableau stabilization was originally developed to construct the sufficiently large rectangular tableaux fixed by given powers of promotion, the purpose of this paper is to improve the original bound on tableau stabilization to the number of rows of the skew tableau. In order to prove this bound, we encode increasing subsequences as lattice paths and show that various operations on these lattice paths weakly increase the maximum combined length of the increasing subsequences.

Keywords

cyclic sieving, promotion, jeu-de-taquin, rectificaton, tableaux, stabilization, permutation

2010 Mathematics Subject Classification

Primary 05E10. Secondary 05E18.

Full Text (PDF format)

Received 5 November 2020

Accepted 19 January 2021

Published 31 January 2022