Journal of Integer Sequences, Vol. 15 (2012), Article 12.6.2

Avoiding Colored Partitions of Two Elements in the Pattern Sense

Adam M. Goyt
Department of Mathematics
Minnesota State University Moorhead
Moorhead, MN 56563

Lara K. Pudwell
Department of Mathematics and Computer Science
Valparaiso University
Valparaiso, IN 46383


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.

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(Concerned with sequences A000027 A000079 A000110 A000898 A000918 A001861 A005425 A005843 A011965 A014322 A052889 A052944 A081124.)

Received March 17 2012; revised version received June 9 2012. Published in Journal of Integer Sequences, June 13 2012.

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