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Avoiding Colored Partitions of Two Elements
in the Pattern Sense Adam M. Goyt
Department of Mathematics
Minnesota State University Moorhead
Moorhead, MN 56563
USA
mailto:goytadam@mnstate.edugoytadam@mnstate.edu

Lara K. Pudwell
Department of Mathematics and Computer Science
Valparaiso University
Valparaiso, IN 46383
USA
mailto:Lara.Pudwell@valpo.eduLara.Pudwell@valpo.edu

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.