Wilf classification of triples of 4-letter patterns I

David Callan, Toufik Mansour, Mark Shattuck

Research output: Contribution to journalArticlepeer-review


This is the first of two papers in which we determine all 242 Wilf classes of triples of 4-letter permutation patterns by showing that there are 32 non-singleton Wilf classes. There are 317 symmetry classes of triples of 4-letter patterns and after computer calculation of initial terms, the problem reduces to showing that counting sequences that appear to be the same (i.e., agree in the first 16 terms) are in fact identical. This amounts to counting avoiders for 107 representative triples. The insertion encoding algorithm (INSENC) applies to many of them and some others have been previously counted. Thus there remain 36 triples. In this paper, we find the generating function for the first 18 of these triples and in a second paper, we treat the other 18. The generating function turns out to be algebraic in each case. Our methods are both combinatorial and analytic, including decompositions by left-right maxima and by initial letters. Sometimes this leads to an algebraic equation for the generating function, sometimes to a functional equation or a multi-index recurrence that succumbs to the kernel method. A bijection is used in one of the cases (Case 50).

Original languageEnglish
Article number5
JournalDiscrete Mathematics and Theoretical Computer Science
Issue number1
StatePublished - 2017

Bibliographical note

Publisher Copyright:
© 2017 by the author(s).


  • Insertion encoding algorithm
  • Kernel method
  • Pattern avoidance
  • Wilf equivalence

ASJC Scopus subject areas

  • Theoretical Computer Science
  • General Computer Science
  • Discrete Mathematics and Combinatorics


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