Stack-Sorting with Consecutive-Pattern-Avoiding Stacks

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.