Stack-Sorting with Consecutive-Pattern-Avoiding Stacks

TL;DR

This paper introduces and analyzes new stack-sorting maps avoiding consecutive patterns, characterizing their properties, enumerating sortable permutations, and exploring their dynamical behavior, including periodic points and iteration bounds.

Contribution

It generalizes existing stack-sorting maps to consecutive-pattern-avoiding variants, characterizes when their sortable sets form permutation classes, and studies their dynamical properties and fixed points.

Findings

Characterized patterns for permutation class formation.

Enumerated sortable permutations for specific patterns.

Determined maximum iterations to reach periodic points.

Abstract

We introduce consecutive-pattern-avoiding stack-sorting maps $\text{SC}_\sigma$, which are natural generalizations of West's stack-sorting map $s$ and natural analogues of the classical-pattern-avoiding stack-sorting maps $s_\sigma$ recently introduced by Cerbai, Claesson, and Ferrari. We characterize the patterns $\sigma$ such that $\text{Sort}(\text{SC}_\sigma)$, the set of permutations that are sortable via the map $s\circ\text{SC}_\sigma$, is a permutation class, and we enumerate the sets $\text{Sort}(\text{SC}_{\sigma})$ for $\sigma\in\{123,132,321\}$. We also study the maps $\text{SC}_\sigma$ from a dynamical point of view, characterizing the periodic points of $\text{SC}_\sigma$ for all $\sigma\in S_3$ and computing $\max_{\pi\in S_n}|\text{SC}_\sigma^{-1}(\pi)|$ for all $\sigma\in\{132,213,231,312\}$. In addition, we characterize the periodic points of the classical-pattern-avoiding stack-sorting map $s_{132}$, and we show that the maximum number of iterations of $s_{132}$ needed to send a permutation in $S_n$ to a periodic point is $n-1$. The paper ends with numerous open problems and conjectures.