The Order of the (123, 132)-Avoiding Stack Sort

TL;DR

This paper investigates the order of permutations under a generalized stack-sorting map avoiding specific patterns, introducing minimally-sorted permutations and establishing their order, thereby extending previous classifications of periodic points.

Contribution

It introduces minimally-sorted permutations as a new concept and determines their order under the (123, 132)-avoiding stack-sorting map, extending prior work on periodic points.

Findings

The order of the (123, 132)-avoiding stack-sorting map on permutations of size n is 2 * floor((n-1)/2).

Introduces the concept of minimally-sorted permutations as an antithesis to highly-sorted permutations.

Strengthens Berlow's 2021 classification of periodic points.

Abstract

Let $s$ be West's deterministic stack-sorting map. A well-known result (West) is that any length $n$ permutation can be sorted with $n-1$ iterations of $s.$ In 2020, Defant introduced the notion of highly-sorted permutations -- permutations in $s^t(S_n)$ for $t \lessapprox n-1.$ In 2023, Choi and Choi extended this notion to generalized stack-sorting maps $s_{\sigma},$ where we relax the condition of becoming sorted to the analogous condition of becoming periodic with respect to $s_{\sigma}.$ In this work, we introduce the notion of minimally-sorted permutations $\mathfrak{M}_n$ as an antithesis to Defant's highly-sorted permutations, and show that $\text{ord}_{s_{123, 132}}(S_n) = 2 \lfloor \frac{n-1}{2} \rfloor,$ strengthening Berlow's 2021 classification of periodic points.