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\vskip 1cm{\LARGE\bf Avoiding Colored Partitions of Two Elements\\
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in the Pattern Sense}
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\large
Adam M. Goyt\\
Department of Mathematics\\
Minnesota State University Moorhead\\
Moorhead, MN 56563 \\
USA \\
\href{mailto:goytadam@mnstate.edu}{\tt goytadam@mnstate.edu}\\
\ \\
Lara K. Pudwell \\
Department of Mathematics and Computer Science\\
Valparaiso University\\
Valparaiso, IN 46383 \\
USA\\
\href{mailto:Lara.Pudwell@valpo.edu}{\tt Lara.Pudwell@valpo.edu}\\
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\begin{abstract}
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 $[k]=\{1,2,\dots,k\}$.  Let
$\Pi_n\wr C_k$ be the set of partitions of $[n]$ with colors from
$[k]$.

In an earlier work, the authors studied 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.
\end{abstract}


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