On consecutive sums in permutations

TL;DR

This paper investigates the number of distinct consecutive sums in permutations, showing that for a random permutation, this number is approximately a specific fraction of n^2, and provides bounds on the maximum possible number of such sums.

Contribution

It answers an old question by Erdős and Harzheim about the typical number of sums and establishes bounds on the maximum number of distinct sums in permutations.

Findings

Random permutations have about (1+e^{-2})/4 * n^2 sums with high probability.

Bounds are provided for the maximum number of distinct sums across all permutations.

Open questions are posed regarding the minimal number of distinct sums.

Abstract

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.