Permutations with small maximal $k$-consecutive sums

TL;DR

This paper determines the exact minimal maximum deviation of $k$-consecutive sums from their average in permutations of integers 1 through n, for specific cases of n and k.

Contribution

It provides exact values of the minimal maximum $k$-consecutive sum deviation for certain n and k, extending understanding of permutation sum properties.

Findings

Exact values of msum(n,k) for specific n and k.

Explicit formulas for msum(n,3), msum(n,4), and msum(n,6).

Enhanced understanding of sum distribution in permutations.

Abstract

Let $n$ and $k$ be positive integers with $n>k$. Given a permutation $(\pi_1,\ldots,\pi_n)$ of integers $1,\ldots,n$, we consider $k$-consecutive sums of $\pi$, i.e., $s_i:=\sum_{j=0}^{k-1}\pi_{i+j}$ for $i=1,\ldots,n$, where we let $\pi_{n+j}=\pi_j$. What we want to do in this paper is to know the exact value of $$\mathrm{msum}(n,k):=\min\left\{\max\{s_i : i=1,\ldots,n\} -\frac{k(n+1)}{2}: \pi \in S_n\right\},$$ where $S_n$ denotes the set of all permutations of $1,\ldots,n$. In this paper, we determine the exact values of $\mathrm{msum}(n,k)$ for some particular cases of $n$ and $k$. As a corollary of the results, we obtain $\mathrm{msum}(n,3)$, $\mathrm{msum}(n,4)$ and $\mathrm{msum}(n,6)$ for any $n$.