Asymptotics of Pattern Avoidance in the Klazar Set Partition and Permutation-Tuple Settings

Definition. A set partition is a partition of the set [n] for some n into any number of nonempty sets, where the order within the partition is irrelevant. We call these sets blocks of the partition. The number of set partitions of [n] is the Bell number Bn. Often, when writing specific set partitions, we will write the partition [n] = S1 ∪ · · · ∪ Sk as S1/ · · · /Sk, where the Si are in increasing order of smallest element, and are written as strings of numbers from least to greatest; for example, [5] = {2, 4} ∪ {1, 3, 5} would be written 135/24. To carry over our notions of avoidance, we define pattern containment on set partitions.