Patterns in colored circular permutations

Cyclic Pattern Containment and Avoidance

“…Of course, the cyclic Erdős-Szekeres Theorem, Theorem 2.1 above, is one such result. There is also a paper of Gray, Lanning and Wang [GLW18] where the authors consider cyclic packing (maximizing the number of copies of a given pattern among all the permutations [σ] ∈ [S n ] for some n) and superpatterns (permutations containing all the patterns [π] ∈ [S k ] for some k). It would be interesting to see if there are nice enumerative formulas for classes consisting of cyclic patterns of length 5 and up.…”

“…We also use square brackets to denote cyclic analogues of objects defined in the linear case. For example, [S n ] is the set of all cyclic permutations of length n. We say a cyclic [GLW18] and patterns in colored cyclic permutations [GLW19].…”

Pattern avoidance of $[4,k]$-pairs in circular permutations

“…Using the above results, we prove that there are 14 Wilf equivalence classes of [4,5]pairs by enumerating their avoidance classes. The OEIS [14] sequences that come up in this enumeration are listed in Table 1.…”

“…Callan [2] and Vella [16] independently studied circular permutations avoiding a fixed pattern of size 4. Gray, Lanning and Wang continued work in this direction and studied other notions of pattern avoidance in circular permutations (see [5,6]). Very recently, Domagalski et al [3] Motivated by the study of pattern avoidance of (3, k)-pairs in set partitions done in [7], we study circular permutations avoiding two patterns {[σ], [τ ]}, where [σ] is of size 4 and [τ ] is of size k. For simplicity, we say that such pairs of patterns are [4, k]-pairs.…”

“…Based on the circular permutation of size 4, we divide our results into three sections. At the end of each section, we also enumerate avoidance classes for [4,5]-pairs.…”