Counting occurrences of subword patterns in non-crossing partitions

Toufik Mansour, Mark Shattuck

Research output: Contribution to journalArticlepeer-review


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 12m and 213m 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.

Original languageEnglish
Article numberP3.03
JournalArt of Discrete and Applied Mathematics
Issue number3
StatePublished - 2023

Bibliographical note

Publisher Copyright:
© 2023 Dominic O'Key.


  • Non-crossing partition
  • combinatorial statistic
  • set partition
  • subword pattern

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics
  • Applied Mathematics


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