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

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.