Pattern occurrences in k-ary words revisited: A few new and old observations

Toufik Mansour, Reza Rastegar

Research output: Contribution to journalArticlepeer-review


In this paper, we study the pattern occurrence in k-ary words. We prove an explicit upper bound on the number of k-ary words avoiding any given pattern using a random walk argument. Additionally, we reproduce one already known result on the exponential rate of growth of pattern occurrence in words and establish a simple connection among pattern occurrences in permutations and k-ary words. A simple yet interesting consequence of this connection is that the Wilf-equivalence of two patterns in words implies their Wilf-equivalence in permutations.

Original languageEnglish
Article number105596
JournalJournal of Combinatorial Theory. Series A
StatePublished - May 2022

Bibliographical note

Funding Information:
R.R. would like to thank Alex Roitershtein for many fruitful conversations on the pattern avoidance and occurrence. We also would like to thank Zachary Hunter for pointing out a gap in the proof of Theorem 1.1 in an earlier draft of this paper.

Publisher Copyright:
© 2022 Elsevier Inc.


  • Pattern avoidance and occurrence
  • Permutations
  • Random walk
  • k-ary words

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics


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