On the maximal sum of the entries of a matrix power

TL;DR

This paper investigates the maximum sum of entries in the square of a matrix with unique entries from 1 to n^2, establishing that this maximum grows on the order of n^7 and providing precise bounds.

Contribution

It proves that the maximal sum of entries of A^2 for such matrices is asymptotically proportional to n^7, with explicit upper and lower bounds.

Findings

The maximal sum p_n grows as Θ(n^7).

Explicit bounds for p_n are provided.

The growth rate is established with precise asymptotic behavior.

Abstract

Let $p_n$ be the maximal sum of the entries of $A^2$, where $A$ is a square matrix of size $n$, consisting of the numbers $1,2,\ldots,n^2$, each appearing exactly once. We prove that $m_n=\Theta(n^7)$. More precisely, we show that $n(240n^{6}+28n^{5}+364n^{4}+210n^{2}-28n+26-105((-1)^{n}+1))/840\leq p_n\leq n^{3}(n^{2}+1)(7n^{2}+5)/24$.