Sorting permutations with pattern-avoiding machines

TL;DR

This paper introduces pattern-avoiding machines, a new type of sorting device using two stacks with pattern constraints, to better understand the complex problem of sorting permutations with two stacks in series.

Contribution

It defines and studies a novel family of sorting devices with pattern constraints, advancing the understanding of permutation sorting with multiple stacks.

Findings

Analysis of pattern-avoiding machine behavior

Insights into sorting permutations with two stacks

Potential framework for solving open combinatorics problems

Abstract

In this work of thesis we introduce and study a new family of sorting devices, which we call pattern-avoiding machines. They consist of two stacks in series, equipped with a greedy procedure. On both stacks we impose a static constraint in terms of pattern containment: reading the content from top to bottom, the first stack is not allowed to contain occurrences of a given pattern $\sigma$, whereas the second one is not allowed to contain occurrences of $21$. By analyzing the behavior of pattern-avoiding machines, we aim to gain a better understanding of the problem of sorting permutations with two consecutive stacks, which is currently one of the most challenging open problems in combinatorics.