On consecutive sums in permutations

We study the number of values taken by the sums ${\sum }_{i=u}^{v-1}{a}_i$, where $a_1,a_2,\dots ,a_n$ is a permutation of $1,2,\dots ,n$ and $1 \leq u < v \leq n+1$. In particular, we show that for a random choice of a permutation, with high probability there are $(\frac{1+e^{-2}}{4}+o(1))n^2$ such sums. This answers an old question of Erd\H{o}s and Harzheim. We also obtain non-trivial bounds on the maximum possible number of distinct sums, ranging over all permutations of $1,2,\dots ,n$. We close with some questions concerning the minimal possible number of distinct sums.