Avoiding Colored Partitions of Lengths Two and Three

TL;DR

This paper explores pattern avoidance in colored set partitions, extending classical concepts to a new enumerative context and connecting it with recent studies on colored permutations and partitions.

Contribution

It provides initial results and a catalog of pattern avoidance in colored partitions, bridging existing work on signed partitions and new enumerative approaches.

Findings

Initial results for pattern avoidance in colored partitions

Connections between colored permutations and set partitions

Extension of noncrossing partition concepts

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.