Cyclic pattern containment and avoidance

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