The 26 Wilf-equivalence classes of length five quasi-consecutive patterns

TL;DR

This paper classifies all quasi-consecutive patterns of length five into 26 Wilf-equivalence classes, providing new proofs for known classifications and expanding understanding of pattern equivalences in combinatorics.

Contribution

It introduces two families of Wilf-equivalences for vincular patterns and explicitly classifies all length five quasi-consecutive patterns into 26 classes.

Findings

Explicit classification of all length five quasi-consecutive patterns into 26 Wilf-equivalence classes.

New proofs for the classification of length 4 and 5 consecutive patterns.

Identification of two families of Wilf-equivalences for vincular patterns.

Abstract

We present two families of Wilf-equivalences for consecutive and quasi-consecutive vincular patterns. These give new proofs of the classification of consecutive patterns of length $4$ and $5$. We then prove additional equivalences to explicitly classify all quasi-consecutive patterns of length $5$ into 26 Wilf-equivalence classes.