Avoidance of vincular patterns by Catalan words

Let C_n denote the set of words (Formula present) on the alphabet of positive integers satisfying (Formula present). The members of 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 C_n has been an ongoing object of study, and members of 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.