  Enumeration of pattern-avoiding objects is an active area of study with
connections to such disparate regions of mathematics as Schubert varieties and
stack-sortable sequences. Recent research in this area has brought attention to
colored permutations and colored set partitions. A colored partition of a set
$S$ is a partition of $S$ with each element receiving a color from the set
$[k]=\{1,2,...,k\}$. Let $\Pi_n\wr C_k$ be the set of partitions of $[n]$ with
colors from $[k]$.
  In an earlier work, the authors study pattern avoidance in colored set
partitions in the equality sense. Here we study pattern avoidance in colored
partitions in the pattern sense. We say that $\sigma\in\Pi_n\wr C_k$ contains
$\pi\in \Pi_m\wr C_\ell$ in the pattern sense if $\sigma$ contains a copy $\pi$
when the colors are ignored and the colors on this copy of $\pi$ are order
isomorphic to the colors on $\pi$. Otherwise we say that $\sigma$ avoids $\pi$.
  We focus on patterns from $\Pi_2\wr C_2$ and find that many familiar and some
new integer sequences appear. We provide bijective proofs wherever possible,
and we provide formulas for computing those sequences that are new.
