Avoiding Colored Partitions of Lengths Two and Three

Abstract

Pattern avoidance in the symmetric group $S_n$ has provided a number of useful connections between seemingly unrelated problems from stack-sorting to Schubert varieties. Recent work has generalized these results to $S_n\wr C_c$, the objects of which can be viewed as "colored permutations". Another body of research that has grown from the study of pattern avoidance in permutations is pattern avoidance in $\Pi_n$, the set of set partitions of $[n]$. Pattern avoidance in set partitions is a generalization of the well-studied notion of noncrossing partitions. Motivated by recent results in pattern avoidance in $S_n \wr C_c$ we provide a catalog of initial results for pattern avoidance in colored partitions, $\Pi_n \wr C_c$. We note that colored set partitions are not a completely new concept. \emph{Signed} (2-colored) set partitions appear in the work of Bj\"{o}rner and Wachs involving the homology of partition lattices. However, we seek to study these objects in a new enumerative context.