Permutations sortable by two stacks in series

TL;DR

This paper investigates the enumeration of permutations sortable by two stacks in series, providing exact counts for small sizes and asymptotic analysis of the generating function's behavior.

Contribution

It introduces a method to count sortable permutations exactly for small sizes and estimates the asymptotic form of their generating function.

Findings

Exact counts for permutations of length less than 20

Approximate coefficients for larger sizes

Asymptotic behavior of the generating function

Abstract

We address the problem of the number of permutations that can be sorted by two stacks in series. We do this by first counting all such permutations of length less than 20 exactly, then using a numerical technique to obtain nineteen further coefficients approximately. Analysing these coefficients by a variety of methods we conclude that the OGF behaves as $$S(z) \sim A (1 - \mu \cdot z)^\gamma,$$ where $\mu =12.45 \pm 0.15,$ $\gamma= 1.5 \pm 0.3,$ and $A \approx 0.02$.