  We consider asymptotics of set partition pattern avoidance in the sense of
Klazar. Our main result derives the asymptotics of the number of set partitions
avoiding a given set partition within an exponential factor, which leads to a
classification of possible growth rates of set partition pattern classes. We
further define a notion of permutation-tuple avoidance, which generalizes
notions of Aldred et al. and the usual permutation pattern setting, and
similarly determine the number of permutation-tuples avoiding a given tuple to
within an exponential factor.
