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In this undergraduate thesis, we expand on the study of statistics onrestricted growth functions avoiding patterns initiated by Campbell, et. al.Restricted growth functions are of interest because they are in bijection withset partitions. We examine the case when a restricted growth function containsa pattern exactly $k$ times, where $k=0$ corresponds to pattern avoidance. Weprove 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 alsoexamine a poset related to $M(n)$, the poset of integer partitions intodistinct parts, and give a conjecture on the new poset's unimodality. We beginwith a brief history of the field and end with a list of conjectures.

Combinatorial statistics on restricted growth functions containing a pattern exactly $k$ times