On consecutive sums in permutations

We study the number of values taken by the sums ₈=ₔ^v-1 aᵢ, where a₁, a₂, , aₙ is a permutation of 1, 2, , n and 1 u<v n+1. In particular, we show that for a random choice of a permutation, with high probability there are (1+e^-24 +o (1) ) n² such sums. This answers an old question of Erdos and Harzheim. We also obtain non-trivial bounds on the maximum possible number of distinct sums, ranging over all permutations of 1, 2, , n. We close with some questions concerning the minimal possible number of distinct sums.