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Avoiding colored partitions of two elements in the pattern sense

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 $Π_n\wr C_k$ 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 $σ\inΠ_n\wr C_k$ contains $π\in Π_m\wr C_\ell$ in the pattern sense if $σ$ contains a copy $π$ when the colors are ignored and the colors on this copy of $π$ are order isomorphic to the colors on $π$. Otherwise we say that $σ$ avoids $π$. We focus on patterns from $Π_2\wr C_2$ 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.