New Wilf-equivalence results for dashed patterns

TL;DR

This paper establishes new Wilf-equivalence results for dashed patterns, providing a unified approach to several open problems and conjectures in pattern avoidance across words, compositions, and permutations.

Contribution

It introduces a sufficient condition for Wilf-equivalence of certain dashed patterns and explains the equidistribution of key parameters on ordered set partitions.

Findings

Unified solution to multiple Wilf-equivalence problems

Verification of a conjecture on permutation Wilf-equivalence

Explanation of parameter equidistribution on ordered set partitions

Abstract

We give a sufficient condition for the two dashed patterns $\tau^{(1)}-\tau^{(2)}-\cdots-\tau^{(\ell)}$ and $\tau^{(\ell)}-\tau^{(\ell-1)}-\cdots-\tau^{(1)}$ to be (strongly) Wilf-equivalent. This permits to solve in a unified way several problems of Heubach and Mansour on Wilf-equivalences on words and compositions, as well as a conjecture of Baxter and Pudwell on Wilf-equivalences on permutations. We also give a better explanation of the equidistribution of the parameters $\MAK+\bMAJ$ and $\MAK'+\bMAJ$ on ordered set partitions. These results can be viewed as consequences of a simple proposition which states that the set valued statistics "descent set'' and "rise set'' are equidistributed over each equivalence class of the partially commutative monoid generated by a poset $(X,\leq)$.