Enumeration and Distribution of Consecutive Equi-$n$-Squares and Their Algebraic Structure
Authors
Andrew Pendleton
Abstract
We introduce consecutive equi-$n$-squares, a variant of equi-$n$-squares in which at least one row or column is in consecutive or reverse-consecutive order, but every element may not appear in every row or column. 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 $n\to\infty$.
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 $n$.
