Avoidance of vincular patterns by Catalan words

Let Cₙ denote the set of words w=w₁ wₙ on the alphabet of positive integers satisfying w₈+₁ wᵢ+1 for 1 i n-1 with w₁=1. The members of Cₙ are known as Catalan words and are enumerated by the n-th Catalan number Cₙ. The problem of finding the cardinality of various avoidance classes of Cₙ has been an ongoing object of study, and members of Cₙ avoiding one or two classical or a single consecutive pattern have been enumerated. In this paper, we extend these results to vincular patterns and seek to determine the cardinality of each avoidance class corresponding to a pattern of type (1, 2) or (2, 1). In several instances, a simple explicit formula for this cardinality may be given. In the more difficult cases, we find only a formula for the (ordinary) generating function which enumerates the class in question. We make extensive use of functional equations in establishing our generating function results.