Counting subwords in flattened partitions of sets

Toufik Mansour, Mark Shattuck, Stephan Wagner

Research output: Contribution to journalArticlepeer-review


In this paper, we consider the problem of avoidance of subword patterns in flattened partitions, which extends recent work of Callan. We determine in all cases explicit formulas and/or generating functions for the number of set partitions of size n which avoid a single subword pattern of length three. The asymptotic behavior of the resulting counting sequences turns out to depend quite heavily on the specific pattern. For the cases of 312 and 213, we make use of the kernel method to determine the generating function which counts the members of the avoidance class. Furthermore, in the cases of 132, 231, and 123, we also find formulas concerning the distribution on the set of partitions for the statistics recording the number of occurrences of the pattern in question and some related bijective proofs are given. Finally, in each of these cases, it is shown that the number of occurrences of the pattern asymptotically follows a normal distribution.

Original languageEnglish
Pages (from-to)1989-2005
Number of pages17
JournalDiscrete Mathematics
Issue number11
StatePublished - 6 Jun 2015

Bibliographical note

Publisher Copyright:
© 2015 Elsevier B.V.


  • Asymptotic enumeration
  • Flattened set partitions
  • Kernel method
  • Pattern avoidance
  • Subword patterns

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics


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