  A set partition $\sigma$ of $[n]=\{1,\cdots ,n\}$ contains another set
partition $\omega$ if a standardized restriction of $\sigma$ to a subset
$S\subseteq[n]$ is equivalent to $\omega$. Otherwise, $\sigma$ avoids $\omega$.
Sagan and Goyt have determined the cardinality of the avoidance classes for all
sets of patterns on partitions of $[3]$. Additionally, there is a bijection
between the set partitions and restricted growth functions (RGFs). Wachs and
White defined four fundamental statistics on those RGFs. Sagan, Dahlberg,
Dorward, Gerhard, Grubb, Purcell, and Reppuhn consider the distributions of
these statistics over various avoidance classes and they obtained four variate
analogues of the previously cited cardinality results. They did the first
thorough study of these distributions. The analogues of their many results
follows for set partitions with exactly $k$ blocks for a specified positive
integer $k$. These analogues are discussed in this work.
