Pattern avoidance and dominating compositions

TL;DR

This paper determines the exact number of Wilf-equivalence classes for pairs of patterns involving set partitions of size 3 with at least two blocks, resolving a previously posed open problem.

Contribution

It proves that the previously known upper bound is tight, establishing the precise count of Wilf-equivalence classes for these pattern pairs.

Findings

The upper bound for the number of Wilf-equivalence classes is exact.

The paper confirms the conjecture posed by Jelínek, Mansour, and Shattuck.

It advances understanding of pattern avoidance in set partitions.

Abstract

Jel\'inek, Mansour, and Shattuck studied Wilf-equivalence among pairs of patterns of the form $\{\sigma,\tau\}$ where $\sigma$ is a set partition of size $3$ with at least two blocks. They obtained an upper bound for the number of Wilf-equivalence classes for such pairs. We show that their upper bound is the exact number of equivalence classes, thus solving a problem posed by them.