Computational Approach to the $SC_{231}$ Consecutive-Pattern-Avoiding Stack Sort

TL;DR

This paper investigates the properties and growth rates of the $SC_{231}$ consecutive-pattern-avoiding stack sort map, providing computational data and mathematical bounds for permutation sort-numbers.

Contribution

It computes sort-numbers for permutations up to length 14, estimates averages up to 1000, and establishes new mathematical bounds for the maximum sort-numbers.

Findings

Sort-numbers grow faster than linear for tested ranges.

Maximum and average sort-numbers increase with permutation length.

Mathematical bounds for maximum sort-numbers are established.

Abstract

Defant and Zheng introduced a consecutive-pattern-avoiding stack sort map $SC_{\sigma}$, where the stack must avoid a consecutive pattern $\sigma$. Seidel and Sun disproved a conjecture in Defant and Zheng's paper about the maximum sort-number of a length $n$ permutation under $SC_{231}$. In this paper, we compute sort-numbers for each permutation of length up to $14$, and we estimate the average sort-numbers up to length $1000$. Our results suggest the maximum and average sort-numbers grow faster than linear with respect to $n$ for the tested ranges, though the long-term behavior remains unclear. We also prove properties of $SC_{231}$ mathematically, such as a $n-1$ lower bound and a $\frac{(n+1)(n-2)}{2}$ upper bound for the maximum sort-number of length $n$ permutations.