The expected variation of random bounded integer sequences of finite length

its most compelling aspect is its vertical variation, that is, the sum of the vertical distances between its adjacent terms. Denoted by varw, the vertical variation of the sequence in (1.1) is varw = 2 + 1 + 0 + 2 + 1 = 6. Our purpose here is to compute the mean and variance of var on four classical sets of combinatorial sequences. To formalize matters and place our problem in the context of other work, let [m]n denote the set of sequences w = x1x2 ···xn of length n with each xi ∈ {1,2, . . . ,m}. For a real-valued function f on [m]2, the f -adjacency number of w = x1x2 ···xn ∈ [m]n is defined to be