Cyclic permutations of sequences and uniform partitions

Let r = (r_i)_i=1ⁿ be a sequence of real numbers of length n with sum s. Let s₀ = 0 and s_i = r₁ +. . .+ r_i for every i ∈ {1, 2,. . ., n}. Fluctuation theory is the name given to that part of probability theory which deals with the fluctuations of the partial sums s_i. Define p(r) to be the number of positive sum si among s₁,. . ., s_n and m(r) to be the smallest index i with s_i = max s_k. An important problem in fluctuation theory is that of showing that in a random path the number of steps on the positive half-line has the same distribution as the index where the maximum is attained for the first time. In this paper, let r_i = (r_i,. . ., r_n, r₁,. . ., r_i-1) be the i-th cyclic permutation of r. For s > 0, we give the necessary and sufficient conditions for {m(r_i) | 1 ≤ i ≤ n} = {1, 2,. . ., n} and {p(r_i) | 1 ≤ i ≤ n} = {1, 2,. . ., n}; for s ≤ 0, we give the necessary and sufficient conditions for {m(r_i) | 1 ≤ i ≤ n} = {0, 1,..., n - 1} and {p(r_i) | 1 ≤ i ≤ n} = {0, 1,. . ., n-1}. We also give an analogous result for the class of all permutations of r.