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

Recently, the general problem of enumerating permutations π = π₁ · · · π_n such that π_i+r − π_i ≠ s for all 1 ≤ i ≤ n − r, where r and s are fixed, was considered by Spahn and Zeilberger. In this paper, we consider an analogous problem on k-ary words involving the distribution of the corresponding statistic. Note that for k-ary words, it suffices to consider only the r = 1 case of the aforementioned problem on permutations. Here, we compute for arbitrary s an explicit formula for the ordinary generating function for n ≥ 0 of the distribution of the statistic on k-ary words ρ = ρ₁ · · · ρ_n recording the number of indices i such that ρ_i+1 − ρ_i = s. 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 sequences7cOxreXsgGF3LkfdtM6Bsow6from the OEIS arise as enumerators of certain classes of k-ary words avoiding adjacencies with a prescribed difference. The comparable problem where one tracks indices i such that the absolute difference |a_i+1 −a_i | is a fixed number is also considered on k-ary words and the corresponding generating function may be expressed in terms of Chebyshev polynomials.