Cyclic pattern containment and avoidance

The study of pattern containment and avoidance for linear permutations is a well-established area of enumerative combinatorics. A cyclic permutation is the set of all rotations of a linear permutation. Vella and Callan independently initiated the study of permutation avoidance in cyclic permutations and characterized the avoidance classes for all single permutations of length 4. We continue this work. In particular, we derive results about avoidance of multiple patterns of length 4, and we determine generating functions for the cyclic descent statistic on these classes. We also consider consecutive pattern containment, and relate the generating functions for the number of occurrences of certain linear and cyclic patterns. Finally, we end with various open questions and avenues for future research.