Periodic Points of Consecutive-Pattern-Avoiding Stack-Sorting Maps

TL;DR

This paper characterizes the periodic points of a generalized stack-sorting map that avoids a specific permutation pattern consecutively, showing they are exactly the permutations avoiding that pattern and its reverse.

Contribution

It proves a key conjecture by Defant and Zheng, identifying the structure of periodic points in the pattern-avoiding stack-sorting map.

Findings

Periodic points are permutations avoiding σ and its reverse.

Confirms the conjecture about the structure of periodic points.

Provides a complete characterization of these points.

Abstract

West's stack-sorting map involves a stack which avoids the permutation $21$ consecutively. Defant and Zheng extended this to a consecutive-pattern-avoiding stack-sorting map $SC_\sigma$, where the stack must always avoid a given permutation $\sigma$ consecutively. We address one of the main conjectures raised by Defant and Zheng in their dynamical approach to $SC_\sigma$. Specifically, we show that the periodic points of $SC_\sigma$ are precisely the permutations that consecutively avoid $\sigma$ and its reverse.