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

TL;DR

This paper investigates pattern occurrences in k-ary words, providing an explicit upper bound for pattern-avoiding words, revisiting known results, and establishing a link between pattern occurrences in permutations and words.

Contribution

It introduces a new upper bound on pattern-avoiding k-ary words and connects pattern occurrences in permutations with those in words, unifying existing results.

Findings

Derived an explicit upper bound for pattern-avoiding words

Reproduced several known results in pattern avoidance

Established a connection between pattern occurrences in permutations and words

Abstract

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 several already known results and establish a simple connection among pattern occurrences in permutations and $k$-ary words. A simple consequence of this connection is that Wilf-equivalence of two patterns in words implies their Wilf-equivalence in permutations.