Highly Sorted Permutations and Bell Numbers

TL;DR

This paper characterizes the iterated preimages of the stack-sorting map on permutations and reveals that their sizes correspond to Bell numbers, establishing a deep combinatorial connection.

Contribution

It provides a simple characterization of the sets obtained by iterated stack-sorting and links their sizes to Bell numbers, including tight bounds for specific cases.

Findings

The size of the iterated preimage set equals the Bell number $B_m$ under certain conditions.

The restriction $n geq 2m-2$ is shown to be tight with explicit formulas.

A new combinatorial connection between stack-sorting and Bell numbers is established.

Abstract

Let $s$ denote West's stack-sorting map. For all positive integers $m$ and all integers $n\geq 2m-2$, we give a simple characterization of the set $s^{n-m}(S_n)$; as a consequence, we find that $|s^{n-m}(S_n)|$ is the $m^\text{th}$ Bell number $B_m$. We also prove that the restriction $n\geq 2m-2$ is tight by showing that $|s^{m-3}(S_{2m-3})|=B_m+m-2$ for all $m\geq 3$.