Pattern avoidance for set partitions à la Klazar

In 2000 Klazar introduced a new notion of pattern avoidance in the context of set partitions of [n] = {1, . . . , n}. The purpose of the present paper is to undertake a study of the concept of Wilf-equivalence based on Klazar’s notion. We determine all Wilf-equivalences for partitions with exactly two blocks, one of which is a singleton block, and we conjecture that, for n ≥ 4, these are all the Wilf-equivalences except for those arising from complementation. If τ is a partition of [k] and Πn(τ ) denotes the set of all partitions of [n] that avoid τ , we establish inequalities between |Πn(τ1)| and |Πn(τ2)| for several choices of τ1 and τ2, and we prove that if τ2 is the partition of [k] with only one block, then |Πn(τ1)| < |Πn(τ2)| for all n > k and all partitions τ1 of [k] with exactly two blocks. We conjecture that this result holds for all partitions τ1 of [k]. Finally, we enumerate Πn(τ ) for all partitions τ of [4].