Avoidance of Type (1,2) Patterns by Catalan Words

Abstract

A certain subset of the multiset permutations of length n satisfying two restrictions has been recently shown to be enumerated by the Catalan number Cn−1. These sequences have been termed Catalan words and are closely related to the 321-avoiding permutations. Here, we consider the problem of avoidance of patterns of type (1,2) wherein the second and third letters within an occurrence of a pattern are required to be adjacent. We derive in several cases functional equations satisfied by the generating functions enumerating members of the avoidance class which we solve by various methods. In one case, the generating function can be expressed in terms of a sum of reciprocals of Chebyshev polynomials, while in another, in terms of a previously studied q-Bell number. Among the sequences arising as enumerators of avoidance classes are the Motzkin and Fibonacci numbers. In several cases, it is more convenient to consider first the problem of avoidance on the subset of Catalan words whose members have no adjacent letters the same before moving to the larger problem on all Catalan words.