Avoiding colored partitions of two elements in the pattern sense

TL;DR

This paper investigates pattern avoidance in colored set partitions, focusing on patterns of size two with two colors, revealing connections to known integer sequences and introducing new formulas and bijections.

Contribution

It introduces the study of pattern avoidance in colored partitions in the pattern sense, especially for size two patterns with two colors, and provides formulas and bijections for new sequences.

Findings

Many familiar integer sequences appear in the enumeration.

New formulas are provided for sequences not previously characterized.

Bijections are constructed for several pattern avoidance classes.

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 $[k]=\{1,2,...,k\}$. 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.