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Vincular Pattern Avoidance on Cyclic Permutations

Pattern avoidance for permutations has been extensively studied, and has been generalized to vincular patterns, where certain elements can be required to be adjacent. In addition, cyclic permutations, i.e., permutations written in a circle rather than a line, have been frequently studied, including in the context of pattern avoidance. We investigate vincular pattern avoidance on cyclic permutations. In particular, we enumerate many avoidance classes of sets of vincular patterns of length 3, including a complete enumeration for all single patterns of length 3. Further, several of the avoidance classes corresponding to a single vincular pattern of length 4 are enumerated by the Catalan numbers. We then study more generally whether sets of vincular patterns of an arbitrary length $k$ can be avoided for arbitrarily long cyclic permutations, in particular investigating the boundary cases of minimal unavoidable sets and maximal avoidable sets.