Avoidance of vincular patterns by Catalan words

Let $\mathcal{C}_n$ denote the set of words $w=w_1\cdots w_n$ on the alphabet of positive integers satisfying $w_{i+1}\leq w_i+1$ for $1 \leq i \leq n-1$ with $w_1=1$. The members of $\mathcal{C}_n$ are known as Catalan words and are enumerated by the $n$-th Catalan number $C_n$. The problem of finding the cardinality of various avoidance classes of $\mathcal{C}_n$ has been an ongoing object of study, and members of $\mathcal{C}_n$ 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.