  In this undergraduate thesis, we expand on the study of statistics on
restricted growth functions avoiding patterns initiated by Campbell, et. al.
Restricted growth functions are of interest because they are in bijection with
set partitions. We examine the case when a restricted growth function contains
a pattern exactly $k$ times, where $k=0$ corresponds to pattern avoidance. We
prove results for several statistic and pattern combinations in the case when
$k=1$ and define a new type of Wilf equivalence for these patterns. We also
examine a poset related to $M(n)$, the poset of integer partitions into
distinct parts, and give a conjecture on the new poset's unimodality. We begin
with a brief history of the field and end with a list of conjectures.
