Note on packing patterns in colored permutations

Packing patterns in permutations concerns finding the permutation with the maximum number of a prescribed pattern. In 2002, Albert, Atkinson, Handley, Holton and Stromquist showed that there always exists a layered permutation containing the maximum number of a layered pattern among all permutations of length n. Consequently the packing density for all but two (up to equivalence) patterns up to length 4 can be obtained. In this note we consider the analogous question for colored patterns and permutations. By introducing the concept of “colored blocks” we characterize the optimal permutations with the maximum number of a given colored pattern when it contains at most three colored blocks. As examples we apply this characterization to find the optimal permutations of various colored patterns and subsequently obtain their corresponding packing densities.