On consecutive sums in permutations

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