Avoidance of vincular patterns by Catalan words

TL;DR

This paper studies how Catalan words avoid vincular patterns, providing explicit formulas and generating functions for various pattern avoidance classes, extending previous work on classical and consecutive patterns.

Contribution

It extends the enumeration of Catalan words avoiding classical and consecutive patterns to vincular patterns of types (1,2) and (2,1), offering explicit formulas and generating functions.

Findings

Explicit formulas for pattern avoidance cardinalities in several cases

Generating functions for more complex avoidance classes

Extension of pattern avoidance enumeration to vincular patterns

Abstract

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.