$k$-pop stack sortable permutations and $2$-avoidance

TL;DR

This paper characterizes permutations sortable by k passes through a pop stack using finitely many patterns, introducing the new concept of 2-avoidance to refine pattern avoidance definitions.

Contribution

It provides a finite pattern characterization for k-pop stack sortable permutations and introduces the novel concept of 2-avoidance for pattern avoidance.

Findings

Finite pattern characterization for all k in natural numbers.

Introduction of the concept of 2-avoidance.

Answer to a question posed by Claesson and Guðmundsson.

Abstract

We consider permutations sortable by $k$ passes through a deterministic pop stack. We show that for any $k\in\mathbb N$ the set is characterised by finitely many patterns, answering a question of Claesson and Gu{\dh}mundsson. Our characterisation demands a more precise definition than in previous literature of what it means for a permutation to avoid a set of barred and unbarred patterns. We propose a new notion called \emph{$2$-avoidance}.