Arrange the integers 1, 2, …, n on a circle and, for a fixed k≥1, let si be the sum of the k consecutive entries starting at position i (indices taken modulo n). For a circular permutation π, define the range R (π) =maxisi−minisi, and let w (n, k) be the minimum value of R (π) over all circular permutations of 1, …, n. We obtain three structural results. First, we prove the complement symmetry w (n, k) =w (n, n−k). Second, we determine the first nontrivial arithmetic progression case n=2k+1 exactly: w (2k+1, k) =2k2. Third, we determine the structured regime n=k2+1 exactly: w (k2+1, k) =k. The proofs combine averaging lower bounds on the progression n≡1 (modk) with explicit constructions: a parity-sensitive two-block arrangement for n=2k+1 and a k×k array construction for n=k2+1.
Yang et al. (Thu,) studied this question.