Permutations sorted by a finite and an infinite stack in series

TL;DR

This paper investigates the set of permutations sortable by a finite stack of depth at least 3 followed by an infinite stack, demonstrating that this set has an infinite basis and thus identifying the transition point in the complexity of such sorting processes.

Contribution

It proves that the permutations sorted by a finite stack of depth t ≥ 3 and an infinite stack in series have an infinite basis, answering an open question in the field.

Findings

The set of permutations sortable by these stacks has an infinite basis.

The transition point from finite to infinite basis occurs at stack depth t=3.

Constructs an infinite antichain to demonstrate the infinite basis.

Abstract

We prove that the set of permutations sorted by a stack of depth $t \geq 3$ and an infinite stack in series has infinite basis, by constructing an infinite antichain. This answers an open question on identifying the point at which, in a sorting process with two stacks in series, the basis changes from finite to infinite.