Stack Sorting with Increasing and Decreasing Stacks

TL;DR

This paper introduces a generalized stack sorting machine with multiple stacks having alternating order restrictions, analyzes the class of permutations it can sort, and presents an optimal sorting algorithm for the case of two stacks.

Contribution

It extends previous work by analyzing a new multi-stack sorting device with alternating order constraints and provides an explicit basis for the sortable permutations when k=2.

Findings

The set of permutations sortable by the k=2 device has an infinite basis.

An optimal sorting algorithm for the case k=2 is described.

Analysis of greedy sorting procedures yields complete and partial results.

Abstract

We introduce a sorting machine consisting of $k+1$ stacks in series: the first $k$ stacks can only contain elements in decreasing order from top to bottom, while the last one has the opposite restriction. This device generalizes \cite{SM}, which studies the case $k=1$. Here we show that, for $k=2$, the set of sortable permutations is a class with infinite basis, by explicitly finding an antichain of minimal nonsortable permutations. This construction can easily be adapted to each $k \ge 3$. Next we describe an optimal sorting algorithm, again for the case $k=2$. We then analyze two types of left-greedy sorting procedures, obtaining complete results in one case and only some partial results in the other one. We close the paper by discussing a few open questions.