Counting pairs of words according to the number of common rises, levels, and descents

A level (L) is an occurrence of two consecutive equal entries in a word w = w₁w₂ . . . , while a rise (R) or descent (D) occurs when the right or left entry, respectively, is strictly larger. If u = u₁u₂ . . . un and v = v₁v₂ . . . v_n are k-ary words of the same given length and 1 ≤ i ≤ n-1, then there is, for example, an occurrence of LR at index i if u_i = u_i+1 and v_i < v_i+1, and, likewise, for the other possibilities. Similar terminology may be used when discussing ordered d-tuples of k-ary words of length n (the set of which we'll often denote by [k]^nd). In this paper, we consider the problem of enumerating the members of [k]^nd according to the number of occurrences of the pattern ρ, where d ≥ 1 and ρ is any word of length d in the alphabet {L,R,D}. In particular, we find an explicit formula for the generating function counting the members of [k]^nd according to the number of occurrences of the patterns ρ = L_iR^d-i, 0 < i < d, which, by symmetry, is seen to solve the aforementioned problem in its entirety. We also provide simple formulas for the average number of occurrences of ρ within all of the members of [k]^nd, providing both algebraic and combinatorial proofs. Finally, in the case d = 2, we solve the problem above where we also allow for weak rises (which we'll denote by R_w), i.e., indices i such that w_i ≤ w_i+1 in w. Enumerating the cases R_wR_w and RR_w seems to be more difficult, and to do so, we combine the kernel method with the simultaneous use of several recurrences.