Counting k-ary words by number of adjacency differences of a prescribed size

Recently, the general problem of enumerating permutations
such that
for all
, where
and
are fixed, was considered by Spahn and Zeilberger. In this paper, we consider an analogous problem on
-ary words involving the distribution of the corresponding statistic. Note that for
-ary words, it suffices to consider only the
case of the aforementioned problem on permutations. Here, we compute for arbitrary
an explicit formula for the ordinary generating function for
of the distribution of the statistic on
-ary words
recording the number of indices
such that
. This result may then be used to find a comparable formula for finite set partitions with a fixed number of blocks, represented sequentially as restricted growth functions. Further, several sequences from the OEIS arise as enumerators of certain classes of
-ary words avoiding adjacencies with a prescribed difference. The comparable problem where one tracks indices
such that the absolute difference
is a fixed number is also considered on
-ary words and the corresponding generating function may be expressed in terms of Chebyshev polynomials.