Descents and des-Wilf Equivalence of Permutations Avoiding Certain Non-Classical Patterns

TL;DR

This paper investigates the concept of des-Wilf equivalence in permutation pattern avoidance, focusing on non-classical patterns and extending previous work on classical pattern avoidance and statistics preservation.

Contribution

It advances the understanding of des-Wilf equivalence for non-classical patterns, building on prior work on classical pattern avoidance and statistical preservation.

Findings

Identifies conditions for des-Wilf equivalence with non-classical patterns

Extends Wilf equivalence concepts to include permutation statistics

Provides new classifications of pattern avoidance equivalences

Abstract

A frequent topic in the study of pattern avoidance is identifying when two sets of patterns $\Pi, \Pi'$ are Wilf equivalent, that is, when $|\text{Av}_n(\Pi)| = |\text{Av}_n(\Pi')|$ for all $n$. In recent work of Dokos et al. the notion of Wilf equivalence was refined to reflect when avoidance of classical patterns preserves certain statistics. In this article, we continue their work by examining $\text{des}$-Wilf equivalence when avoiding certain non-classical patterns.