Set partition patterns and statistics

A set partition σσ of [n]={1,…,n}[n]={1,…,n} contains another set partition ππ if restricting σσ to some S⊆[n]S⊆[n] and then standardizing the result gives ππ. Otherwise we say σσ avoids ππ. For all sets of patterns consisting of partitions of [3], the sizes of the avoidance classes were determined by Sagan and by Goyt. Set partitions are in bijection with restricted growth functions (RGFs) for which Wachs and White defined four fundamental statistics. We consider the distributions of these statistics over various avoidance classes, thus obtaining multivariate analogues of the previously cited cardinality results. This is the first in-depth study of such distributions. We end with a list of open problems.