a(n, k) is defined to be the number of ways to express n as an ordered sum of k positive integers, up to cyclic permutation. We will prove that for any n and k with 0 < k < n, a(n, k) = a(n, n - k).
Let b(n, k) be the number of ways to express n as an ordered sum of k nonnegative integers, up to cyclic permutation. By adding one to each term in such a sum, we get a sum of the type counted by a(n, k). This proves that for any positive integers n and k, b(n, k) = a(n + k, k). So a(n, k) = b(n - k, k), and a(n, n - k) = b(k, n - k). Therefore it will suffice to prove that b(n, k) = b(k, n) for any n, k > 0.

Let S_nk be the set of ordered sequences of k nonnegative integers with sum n, up to cyclic permutation. We will describe a bijection f_nk: S_nk -> S_kn. Given a sequence a = (a_1, a_2, ..., a_k) with sum n, we draw a path on a rectangular grid as follows: starting at any lattice point, we go a_1 steps up, then 1 step right, then a_2 steps up, then 1 step right, etc., ending with a_k steps up and 1 step right. This brings us to a point k steps right and n steps up from the starting point. We then extend this path infinitely in both directions to make it periodic with period (k, n). As a result of this extension, any cyclic permutation of a yields the same path, up to translation. Observe that we can read a from the path by picking k consecutive vertical gridlines, and counting how far up the path goes along each of these lines. If instead we reverse the roles of horizontal and vertical by picking n consecutive horizontal gridlines, and count how far right the path goes along each one, we get a sequence (b_1, b_2, ..., b_n) with sum k. Let this sequence be f_nk(a). Then f_kn(f_nk(a)) = a, because applying f_kn means again reversing the roles of horizontal and vertical. This proves that for any n, k > 0, f_nk is a bijection, with inverse f_kn.