Wilf collapse in permutation classes

TL;DR

This paper investigates the phenomenon of Wilf collapse in hereditary permutation classes, demonstrating that it is widespread, especially in classes with unbounded growth and finitely many sum-indecomposables, using word encoding techniques.

Contribution

It establishes that Wilf collapse occurs in many permutation classes, providing new insights into their structural and enumerative properties.

Findings

Wilf collapse occurs in classes with unbounded growth

Wilf collapse occurs in classes with finitely many sum-indecomposables

Encoding permutations as words aids in analyzing Wilf phenomena

Abstract

For a hereditary permutation class $\mathcal{C}$, we say that two permutations $\pi$ and $\sigma$ of $\mathcal{C}$ are Wilf-equivalent in $\mathcal{C}$, if $\mathcal{C}$ has the same number of permutations avoiding $\pi$ as those avoiding $\sigma$. We say that a permutation class $\mathcal{C}$ exhibits a Wilf collapse if the number of permutations of size $n$ in $\mathcal{C}$ is asymptotically larger than the number of Wilf-equivalence classes formed by these permutations. In this paper, we show that Wilf collapse is a surprisingly common phenomenon. Among other results, we show that Wilf collapse occurs in any permutation class with unbounded growth and finitely many sum-indecomposable permutations. Our proofs are based on encoding the elements of a permutation class $\mathcal{C}$ as words, and analyzing the structure of a random permutation in $\mathcal{C}$ using this representation.