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.