Enumeration of words by the sum of differences between adjacent letters

and u is the total variation (or just variation) of σ. For example, the total variation of the word σ = 1121415221 ∈ [5] is u(σ) = 0 + 1 + 1 + 3 + 3 + 4 + 3 + 0 + 1 = 16. Several authors have studied the statistic u(σ) in different contexts. For example, Curtis [5] discussed the problem of finding a word σ = σ1σ2 · · ·σn over the alphabet {a1, a2, . . . , ak} which maximizes the risk function ∑n j=1 |σi−σi−1|, where σ0 = σn. Chao and Liang [3] studied the permutations of n distinct letters arranged on a circle or a line and found the arrangements that maximize the risk function. Later, Cohen and Tonkes [4] (see also [5, 9]) analyzed optimal permutations for multisets. Motivated by the above results, we pose the following problem: