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

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.