Stanley-Wilf limits are typically exponential

TL;DR

This paper disproves the conjecture that Stanley-Wilf limits grow quadratically with permutation size, showing instead that for most permutations, these limits grow exponentially with the permutation length.

Contribution

The paper demonstrates that Stanley-Wilf limits are typically exponential, contradicting previous conjectures of quadratic growth for all permutations.

Findings

Stanley-Wilf limits are typically exponential in permutation length.

The quadratic growth conjecture for Stanley-Wilf limits is false for most permutations.

Almost all permutations have Stanley-Wilf limits growing as 2^{k^{Θ(1)}}.

Abstract

For a permutation $\pi$, let $S_{n}(\pi)$ be the number of permutations on $n$ letters avoiding $\pi$. Marcus and Tardos proved the celebrated Stanley-Wilf conjecture that $L(\pi)= \lim_{n \to \infty} S_n(\pi)^{1/n}$ exists and is finite. Backed by numerical evidence, it has been conjectured by many researchers over the years that $L(\pi)=\Theta(k^2)$ for every permutation $\pi$ on $k$ letters. We disprove this conjecture, showing that $L(\pi)=2^{k^{\Theta(1)}}$ for almost all permutations $\pi$ on $k$ letters.