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.
Bibliographical noteFunding 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.
© 2022 Elsevier Inc.
- Pattern avoidance and occurrence
- Random walk
- k-ary words
ASJC Scopus subject areas
- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics