Moments of Magic Squares: Line-Sum Conditions, a Permutation Threshold, and Random Energies

Kyle Elliott Mathewson

Every semi-magic square of order $n$ satisfies the axial moment identities ${\sum }_{i,j}{s}_{ij}{j}^m={\sum }_{i,j}{s}_{ij}{i}^m=M(n){\sum }_j{j}^m$ for all $m\ge 1$, where $M(n)=n(n^2+1)/2$. For rea…