Disorders and Permutations

The additive x-disorder of a permutation is the sum of the absolute differences of all pairs of consecutive elements. We show that the additive x-disorder of a permutation of S(n), n ≥ 2, ranges from n − 1 to ⌊ n2/2 ⌋ − 1, and we give a complete characterization of permutations having extreme such values. Moreover, for any positive integers n and d such that n ≥ 2 and n − 1 ≤ d ≤ ⌊ n2/2 ⌋ − 1, we propose a linear-time algorithm to compute a permutation π ∈ S(n) with additive x-disorder d. 2012 ACM Subject Classification Mathematics of computing → Permutations and combinations