Moments of Magic Squares: Line-Sum Conditions, a Permutation Threshold, and Random Energies
Abstract
Every semi-magic square of order n satisfies the axial moment identities ∑i,jsijjm=∑i,jsijim=M(n)∑jjm for all m≥1, where M(n)=n(n2+1)/2. For real matrices the converse requires 1≤ m≤ n−1, whereas for arrangements of {1,…,n2} fewer moments suffice once n≥14. We define the permutation threshold k*(n) as the least k such that every arrangement satisfying these identities for 1≤ m≤ k is semi-magic. In terms of the column-sum deviations γj, the degree-k conditions state that ∑jγjtj is divisible by (t−1)k+1, linking k*(n) to the Prouhet--Tarry--Escott problem. While Schur's bound on real zeros gives k*(n)≤√(2(n−1)log(n4/4)), we prove the converse realisability bound k*(n)≥(√2−o(1))√n (and k*(n)≥(√6−o(1))√n when 4| n) by perturbing semi-magic squares with complementary row pairs via horizontal swaps. We also show that k*(n)=n−1 for 3≤ n≤13, k*(n)=n−2 for 14≤ n≤17, and k*(n)≤ n−3 for n≥18, and determine k*(n) for 18≤ n≤40 with computer assistance. Finally, at the k=1 level of the hierarchy, we specialise the Mantel--Hubert quadratic assignment variance formula to obtain exact means, variances, and limit laws under a uniform random arrangement for the full line-sum energy Efull, for the four-term energy E4 combining the first moments and the two diagonals, and for Loly's moment of inertia, and find that |L4|=14,639,440 arrangements of order $4$ satisfy E4=0.
Related
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BibTeX
@misc{mathewson2026moments,
title = {{Moments of Magic Squares: Line-Sum Conditions, a Permutation Threshold, and Random Energies}},
author = {Kyle Elliott Mathewson},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.38386},
eprint = {2610.38386},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.38386},
}
RIS
TY - UNPB
TI - Moments of Magic Squares: Line-Sum Conditions, a Permutation Threshold, and Random Energies
AU - Kyle Elliott Mathewson
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.38386
UR - https://arxiv.org/abs/2610.38386
AB - Every semi-magic square of order n satisfies the axial moment identities ∑_i,js_ijj^m=∑_i,js_iji^m=M(n)∑_jj^m for all m≥1, where M(n)=n(n^2+1)/2. For real matrices the converse requires 1≤ m≤ n−1, whereas for arrangements of {1,…,n^2} fewer moments suffice once n≥14. We define the permutation threshold k^*(n) as the least k such that every arrangement satisfying these identities for 1≤ m≤ k is semi-magic. In terms of the column-sum deviations γ_j, the degree-k conditions state that ∑_jγ_jt^j is divisible by (t−1)^k+1, linking k^*(n) to the Prouhet--Tarry--Escott problem. While Schur's bound on real zeros gives k^*(n)≤√(2(n−1)log(n^4/4)), we prove the converse realisability bound k^*(n)≥(√2−o(1))√n (and k^*(n)≥(√6−o(1))√n when 4| n) by perturbing semi-magic squares with complementary row pairs via horizontal swaps. We also show that k^*(n)=n−1 for 3≤ n≤13, k^*(n)=n−2 for 14≤ n≤17, and k^*(n)≤ n−3 for n≥18, and determine k^*(n) for 18≤ n≤40 with computer assistance. Finally, at the k=1 level of the hierarchy, we specialise the Mantel--Hubert quadratic assignment variance formula to obtain exact means, variances, and limit laws under a uniform random arrangement for the full line-sum energy E_full, for the four-term energy E_4 combining the first moments and the two diagonals, and for Loly's moment of inertia, and find that |L_4|=14,639,440 arrangements of order $4$ satisfy E_4=0.
ER -