Passing through a stack $k$ times with reversals

TL;DR

This paper studies a stack sorting process involving multiple passes and reversals, identifying bases for sortable permutations, and establishing connections with Entringer families and alternating permutations.

Contribution

It introduces the basis for 2-reverse pass sortable permutations, proves finiteness of classes for higher rev-tiers, and discovers a new Entringer family linked to alternating permutations.

Findings

Identified the basis for 2-reverse pass sortable permutations.

Proved all classes of (t+1)-reverse pass sortable permutations are finitely based.

Discovered a new Entringer family related to maximal rev-tier permutations.

Abstract

We consider a stack sorting algorithm where only the appropriate output values are popped from the stack and then any remaining entries in the stack are run through the stack in reverse order. We identify the basis for the $2$-reverse pass sortable permutations and give computational results for some classes with larger maximal rev-tier. We also show all classes of $(t+1)$-reverse pass sortable permutations are finitely based. Additionally, a new Entringer family consisting of maximal rev-tier permutations of length $n$ was discovered along with a bijection between this family and the collection of alternating permutations of length $n-1$. We calculate generating functions for the number permutations of length $n$ and exact rev-tier $t$.