Stack-sorting with consecutive-pattern-avoiding stacks

Abstract

We introduce consecutive-pattern-avoiding stack-sorting maps SCσ, which are natural generalizations of West's stack-sorting map s and natural analogues of the classical-pattern-avoiding stack-sorting maps sσ recently introduced by Cerbai, Claesson, and Ferrari. We characterize the patterns σ such that Sort(SCσ), the set of permutations that are sortable via the map s∘SCσ, is a permutation class, and we enumerate the sets Sort(SCσ) for σ∈{123,132,321}. We also study the maps SCσ from a dynamical point of view, characterizing the periodic points of SCσ for all σ∈S3 and computing maxπ∈Sn⁡|SCσ−1(π)| for all σ∈{132,213,231,312}. In addition, we characterize the periodic points of the classical-pattern-avoiding stack-sorting map s132, and we show that the maximum number of iterations of s132 needed to send a permutation in Sn to a periodic point is n−1. The paper ends with numerous open problems and conjectures.