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Enumeration and Distribution of Consecutive Equi-$n$-Squares and Their Algebraic Structure

Andrew Pendleton

TL;DR

The paper introduces consecutive equi-$n$-squares (CE$n$S) and develops a complete counting framework relative to ordinary equi-$n$-squares ($\\Omega_n$). It derives exact and asymptotic formulas for the number of CE$n$Ss, showing their proportion among all equi-$n$-squares vanishes as $n$ grows, and analyzes the distribution of CE$n$Ss under uniform sampling. The work further connects these combinatorial objects to algebraic structures via Cayley tables, proving links to magmas and completely simple semigroups, and validates results with Monte Carlo simulations for small $n$. Together, these results illuminate both the combinatorial rarity of CE$n$Ss and their algebraic significance, with implications for related designs and cryptographic considerations.

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 $n$.

Enumeration and Distribution of Consecutive Equi-$n$-Squares and Their Algebraic Structure

TL;DR

The paper introduces consecutive equi--squares (CES) and develops a complete counting framework relative to ordinary equi--squares (). It derives exact and asymptotic formulas for the number of CESs, showing their proportion among all equi--squares vanishes as grows, and analyzes the distribution of CESs under uniform sampling. The work further connects these combinatorial objects to algebraic structures via Cayley tables, proving links to magmas and completely simple semigroups, and validates results with Monte Carlo simulations for small . Together, these results illuminate both the combinatorial rarity of CESs and their algebraic significance, with implications for related designs and cryptographic considerations.

Abstract

We introduce consecutive equi--squares, a variant of equi--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--squares, showing precisely how their proportion among all equi--squares rapidly approaches zero as . We also analyze the distribution of consecutive equi--squares under uniform random sampling and explore connections to algebraic structures, interpreting equi--squares and consecutive equi--squares as Cayley tables. Finally, we supplement our theoretical results with Monte Carlo simulations for small values of .

Paper Structure

This paper contains 11 sections, 12 theorems, 64 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Lemmas
  3. Proof of Theorem \ref{['main_theorem']}
  4. Proof of Corollary \ref{['asymp_coro']}
  5. Proof of Theorem \ref{['distro_theorem']}
  6. Remark on the Algebra of Consecutive Equi-$n$-Squares
  7. Proof of Theorem \ref{['no QGs are Ls']}
  8. Proof of Theorem \ref{['completely simple']}
  9. Proof of Corollary \ref{['nonLatin_coro']}
  10. Code and Monte-Carlo Results
  11. Acknowledgments

Key Result

Theorem 1.1

The number of consecutive equi-$n$-squares (the number of elements in $\Sigma_n$) is and

Figures (3)

  • Figure 1: Differences between simulated and predicted probabilities for $p_{X_3}(x)$
  • Figure 2: Differences between simulated and predicted probabilities for $p_{X_4}(x)$
  • Figure 3: Differences between simulated and predicted probabilities for $p_{X_5}(x)$

Theorems & Definitions (28)

  • Definition 1.1: Latin Square
  • Definition 1.2: Equi-$n$-Square
  • Definition 1.3: Consecutive Equi-$n$-Square
  • Theorem 1.1
  • Corollary 1.1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.4.1
  • Remark
  • ...and 18 more