Counting occurrences of subword patterns in non-crossing partitions

Abstract

A permutation pattern in which all letters within an occurrence are required to be adjacent is known as a subword. In this paper, we consider the distribution of several infinite families of subword patterns on the set of non-crossing partitions of size n, denoted by NCn, and derive formulas for the generating functions of these distributions on NCn. As special cases of our results, we obtain formulas for the generating functions enumerating the members of NCn according to the number of occurrences of any subword of length three with distinct letters. Simple expressions for the total number of occurrences of a pattern over all members of NCn are also deduced. Some connections are made with the related problem of counting Dyck paths according to the number of occurrences of certain types of strings. Further, for the subwords 12⋯m and 213⋯m where m ≥ 3, we consider the joint distribution with the descents statistic and make use of the kernel method to establish the results in these cases.