Avoiding Colored Partitions of Two Elements in the Pattern Sense

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 Πn ≀ Ck 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 σ ∈ Πn ≀ Ck contains π ∈ Πm ≀ Cl in the pattern sense if σ contains a copy π when the colors are ignored and the colors on this copy of π are order isomorphic to the colors on π. Otherwise we say that σ avoids π. We focus on patterns from Π2 ≀C2 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.