Cyclic Counting Trios

In this paper, we extend the concept of mutually counting sequences discussed in [1] to the case of three sequences of the same length. Specifically, given the positive integer n > 1, we define three sequences, where a(i) is the multiplicity of i in 5, M j) is the multiplicity of j in C, and M &) is the multiplicity of k in ^4. We call the ordered triple (A 9 B 9 C) a cyclic counting trio, and we make some preliminary observations: (i) the entries in sequences A, B, and C are nonnegative integers less than n. (iii) if (A, B 9 C) is a cyclic counting trio, then so are (5, C, A) and (C, A, B). Such permuted trios will not be considered to be different . We say that the cyclic counting trio (A, B 9 C) is redundant if A 9 B, and C are identical. In what follows, we show that there is a unique redundant trio for each n ^ 7': M O) = n-4, M l) = 2, a(2) = 1, a(n-4) = 1, M i) = 0 for all remaining i. There are also two redundant trios when n = 4,. one when n = 5, and no others. Furthermore, we show that a nonredundant trio results only when n = 7: M O) = 4, M l) = 1, a(3) = 2, M 2) = M 4) = M 5) = a(6) = 0; M O) = 3, M l) = 3, M 4) = 1, M 2) = b(3) = i(5) = b(6) = 0; c(0) = 4, M l) = M 2) = M 4) = 1, c(3) = M 5) = M 6) = 0. As a way to become familiar with the problem, we invite the interested reader to investigate the existence of cyclic counting trios when n < 7. We will therefore proceed under the assumption that (A s B 9 C) is a cyclic counting trio and that n > 7. For future reference, we let n* = n-\~1 , 1987] n