Preimages under the Stack-Sorting Algorithm

TL;DR

This paper develops a method to determine preimages under the stack-sorting map, leading to improved upper bounds on the growth rates of t-stack sortable permutations for t=3 and 4.

Contribution

It introduces a recursive approach to bound the number of preimages, enhancing existing bounds for the asymptotic growth of t-stack sortable permutations.

Findings

Significantly improved upper bounds for t=3 and t=4.

Derived recursive bounds for permutation counts.

Enhanced understanding of permutation preimages under stack-sorting.

Abstract

We use a method for determining the number of preimages of any permutation under the stack-sorting map in order to obtain recursive upper bounds for the numbers $W_t(n)$ and $W_t(n,k)$ of $t$-stack sortable permutations of length $n$ and $t$-stack sortable permutations of length $n$ with exactly $k$ descents. From these bounds, we are able to significantly improve the best known upper bounds for $\displaystyle{\lim_{n\to\infty}\sqrt[n]{W_t(n)}}$ when $t=3$ and $t=4$.