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Latin hypercubes realizing integer partitions

Diane Donovan, Tara Kemp, James Lefevre

TL;DR

This work extends Fuchs' 2-realization problem to $m$-ary quasigroups by studying $m$-realizations via latin hypercubes and develops a general toolkit based on outline rectangles and boxes. It provides constructive methods to build latin cubes from latin squares, inflations, and prolongations, and proves that a $2$-RP always yields a $3$-RP, with two key necessary (but not sufficient) conditions guiding the existence of $3$-realizations. The paper delivers detailed results for partitions with at most two distinct parts, establishing sharp bounds such as $a\le 2b$ for $\mathrm{3-RP}(a^1b^2)$ and $a\le (n-1)b$ in higher-part scenarios, while highlighting limits via outline-box non-extendability. Finally, it generalizes to $m$-dimensional hypercubes, showing how realizations can be propagated across dimensions and establishing congruence-based growth, with explicit high-dimensional examples demonstrating richer realizability phenomena beyond the binary case.

Abstract

For an integer partition $h_1 + \dots + h_n = N$, a 2-realization of this partition is a latin square of order $N$ with disjoint subsquares of orders $h_1,\dots,h_n$. The existence of 2-realizations is a partially solved problem posed by Fuchs. In this paper, we extend Fuchs' problem to $m$-ary quasigroups, or, equivalently, latin hypercubes. We construct latin cubes for some partitions with at most two distinct parts and highlight how the new problem is related to the original.

Latin hypercubes realizing integer partitions

TL;DR

This work extends Fuchs' 2-realization problem to -ary quasigroups by studying -realizations via latin hypercubes and develops a general toolkit based on outline rectangles and boxes. It provides constructive methods to build latin cubes from latin squares, inflations, and prolongations, and proves that a -RP always yields a -RP, with two key necessary (but not sufficient) conditions guiding the existence of -realizations. The paper delivers detailed results for partitions with at most two distinct parts, establishing sharp bounds such as for and in higher-part scenarios, while highlighting limits via outline-box non-extendability. Finally, it generalizes to -dimensional hypercubes, showing how realizations can be propagated across dimensions and establishing congruence-based growth, with explicit high-dimensional examples demonstrating richer realizability phenomena beyond the binary case.

Abstract

For an integer partition , a 2-realization of this partition is a latin square of order with disjoint subsquares of orders . The existence of 2-realizations is a partially solved problem posed by Fuchs. In this paper, we extend Fuchs' problem to -ary quasigroups, or, equivalently, latin hypercubes. We construct latin cubes for some partitions with at most two distinct parts and highlight how the new problem is related to the original.
Paper Structure (6 sections, 23 theorems, 11 equations, 11 figures)

This paper contains 6 sections, 23 theorems, 11 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Constructions
  3. Outline Rectangles and Boxes
  4. Some conditions on 3-realizations
  5. Partitions with at most two distinct parts
  6. $m$-dimensional hypercubes

Key Result

Theorem 1.5

Take a partition $(h_1,h_2,\dots,h_n)$ of $N$ with $h_1\geq h_2\geq \dots\geq h_n > 0$. Then a $\mathop{\mathrm{2-RP}}\nolimits(h_1h_2\dots h_n)$

Figures (11)

  • Figure 3:
  • Figure 7:
  • Figure 8:
  • Figure 9:
  • Figure 10: The outline rectangle $O$
  • ...and 6 more figures

Theorems & Definitions (51)

  • Definition 1.1
  • Definition 1.3
  • Definition 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Definition 1.10
  • Definition 1.11
  • ...and 41 more