Patterns in Shi tableaux and Dyck paths

TL;DR

This paper explores pattern containment in Shi tableaux, which are linked to Dyck paths, analyzing their poset structure, pattern avoidance, and providing explicit formulas for covers, with implications for geometric and combinatorial understanding.

Contribution

It introduces a new pattern occurrence notion on Shi tableaux, studies the poset structure, and generalizes pattern avoidance results to larger tableaux, connecting geometric and combinatorial aspects.

Findings

Explicit formulas for upper and lower covers in the Shi tableaux poset

Characterization of pattern avoidance for small tableaux

Generalization of results to larger tableaux sizes

Abstract

Shi tableaux are special binary fillings of certain Young diagrams which arise in the study of Shi hyperplane arrangements related to classical root systems. For type $A$, the set $\mathcal T$ of Shi tableaux naturally coincides with the set of Dyck paths, for which various notions of patterns have been introduced and studied over the years. In this paper we define a notion of pattern occurrence on $\mathcal T$ which, although it can be regarded as a pattern on Dyck paths, it is motivated by the underlying geometric structure of the tableaux. Our main goal in this work is to study the poset of Shi tableaux defined by pattern-containment. More precisely, we determine explicit formulas for upper and lower covers for each $T\in\mathcal T$, we consider pattern avoidance for the smallest non-trivial tableaux (size 2) and generalize these results to certain tableau of larger size. We conclude with open problems and possible future directions.