Enumeration and Distribution of Permutation Rows and Columns in Equi-n-Squares

2025/10/13 by Andrew Pendleton, Pendleton, Andrew
Engineering · Mathematics · #Advanced Mathematical Theories #Combinatorics (math.CO) #FOS: Mathematics #Mathematics and Applications #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2510.11980

openalex publication_date 2025/10/13 · openalex created_date 2025/10/17 · openalex updated_date 2026/08/03

Abstract

We introduce consecutive equi-n-squares, a variant of equi-n-squares in which at least one row or column forms a fixed permutation of \1,…,n\, taken for concreteness to be (1,…,n). More generally, the enumeration and probabilistic arguments presented here extend to the occurrence of any prescribed permutation as a row or column of an equi-n-square. We derive exact and asymptotic formulas for the number of consecutive equi-n-squares, showing precisely how their proportion among all equi-n-squares rapidly approaches zero as n→∞. We also analyze the distribution of consecutive equi-n-squares under uniform random sampling and explore connections to algebraic structures, interpreting equi-n-squares and consecutive equi-n-squares as Cayley tables. Finally, we supplement our theoretical results with Monte Carlo simulations for small values of n.

Citations

• Almost-full transversals in equi-n-squares
• A counterexample to Stein's Equi-n-square Conjecture
• On a conjecture of Stein
• Quantum Latin squares and unitary error bases

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• Shuffled equi-n-squares
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