Counting occurrences of subword patterns in non-crossing partitions

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 subword patterns on set non-crossing partitions size n, denoted by NCn, and derive formulas for generating functions these distributions NCn. As special cases our results, obtain enumerating members NCn according number occurrences any length three with distinct letters. Simple expressions total over also deduced. Some connections made related problem counting Dyck paths certain types strings. Further, subwords 12⋯m 213⋯m where m ≥ 3, joint descents statistic make use kernel method establish results cases.