Two examples of Wilf-collapse

TL;DR

This paper demonstrates exponential Wilf-collapse in two permutation classes, showing that their principal subclasses' enumerations grow much more slowly than expected due to local symmetries and pattern detection algorithms.

Contribution

It introduces two specific classes exhibiting Wilf-collapse and explains the underlying local symmetry mechanism causing this phenomenon.

Findings

Exponential Wilf-collapse observed in X-class and oscillation subclasses.

Local symmetry combined with greedy pattern detection explains the collapse.

Growth of subclasses' enumerations is much slower than the class itself.

Abstract

Two permutation classes, the X-class and subpermutations of the increasing oscillation are shown to exhibit an exponential Wilf-collapse. This means that the number of distinct enumerations of principal subclasses of each of these classes grows much more slowly than the class itself whereas a priori, based only on symmetries of the class, there is no reason to expect this. The underlying cause of the collapse in both cases is the ability to apply some form of local symmetry which, combined with a greedy algorithm for detecting patterns in these classes, yields a Wilf-collapse.