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Sorting permutations using a pop stack with a bypass

Lapo Cioni, Luca Ferrari, Rebecca Smith

TL;DR

The paper introduces a new sorting device, a pop stack with a bypass (PSB), and shows sortable permutations are exactly those avoiding patterns $231$ and $4213$ (i.e., $Av(231,4213)$). It connects the enumeration of this class to the odd-indexed Fibonacci numbers, via a bijection with a restricted Motzkin-path family, and provides an algorithm to compute preimages and to enumerate permutations with few preimages, including descriptions of principal classes. It also discusses a parallel-device variant with bypass. These results advance permutation-sorting theory for constrained containers by giving explicit, constructive characterizations and procedures.

Abstract

We introduce a new sorting device for permutations which makes use of a pop stack augmented with a bypass operation. This results in a sorting machine, which is more powerful than the usual Popstacksort algorithm and seems to have never been investigated previously. In the present paper, we give a characterization of sortable permutations in terms of forbidden patterns and reinterpret the resulting enumerating sequence using a class of restricted Motzkin paths. Moreover, we describe an algorithm to compute the set of all preimages of a given permutation, thanks to which we characterize permutations having a small number of preimages. Finally, we provide a full description of the preimages of principal classes of permutations, and we discuss the device consisting of two pop stacks in parallel, again with a bypass operation.

Sorting permutations using a pop stack with a bypass

TL;DR

The paper introduces a new sorting device, a pop stack with a bypass (PSB), and shows sortable permutations are exactly those avoiding patterns and (i.e., ). It connects the enumeration of this class to the odd-indexed Fibonacci numbers, via a bijection with a restricted Motzkin-path family, and provides an algorithm to compute preimages and to enumerate permutations with few preimages, including descriptions of principal classes. It also discusses a parallel-device variant with bypass. These results advance permutation-sorting theory for constrained containers by giving explicit, constructive characterizations and procedures.

Abstract

We introduce a new sorting device for permutations which makes use of a pop stack augmented with a bypass operation. This results in a sorting machine, which is more powerful than the usual Popstacksort algorithm and seems to have never been investigated previously. In the present paper, we give a characterization of sortable permutations in terms of forbidden patterns and reinterpret the resulting enumerating sequence using a class of restricted Motzkin paths. Moreover, we describe an algorithm to compute the set of all preimages of a given permutation, thanks to which we characterize permutations having a small number of preimages. Finally, we provide a full description of the preimages of principal classes of permutations, and we discuss the device consisting of two pop stacks in parallel, again with a bypass operation.

Paper Structure

This paper contains 3 sections, 1 theorem, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries on permutations and sorting algorithms
  3. An optimal algorithm to sort permutations using a pop stack with a bypass

Key Result

Proposition 3.1

Let $\pi$ be a permutation such that $231\le\pi$ or $4213\le\pi$. Then $\pi$ cannot be sorted using a pop stack with bypass.

Figures (1)

  • Figure :

Theorems & Definitions (2)

  • Proposition 3.1
  • proof