Tableau Stabilization and Lattice Paths

TL;DR

This paper improves bounds on tableau stabilization by encoding increasing subsequences as lattice paths and analyzing their properties to understand when copies of skew tableaux stabilize during rectification.

Contribution

It introduces a new bound on tableau stabilization related to the number of rows, using lattice path encoding to analyze increasing subsequences.

Findings

Bound on tableau stabilization improved to the number of rows

Lattice path encoding of increasing subsequences used in proofs

Operations on lattice paths weakly increase maximum subsequence length

Abstract

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.