Subsquares in random Latin rectangles

TL;DR

This work analyzes the presence of subsquares inside a uniformly random $k\times n$ Latin rectangle using a switching method based on row, column, and symbol cycles. It proves that for all $m$ with $4\le m\le \min\{k,n/2\}$, the expected number of $m\times m$ subsquares is $O(n^{-2})$, and that $\mathbb{E}_3(k,n)=O(k/n)$, while $\mathbb{E}_2(k,n)=(\tfrac12+o(1))\binom{k}{2}$ for all $k\le n$. Consequently, with probability $1-O(1/n)$ there are no proper subsquares of order at least $4$, resolving a conjecture of Divoux, Kelly, Kennedy and Sidhu in the stated regime and strengthening the understanding of subsquare scarcity. The analysis combines large- and small-subsquare bounds, using refined switching arguments to control the exposure of cells and the resulting combinatorial structures. Altogether, the results provide a precise asymptotic picture of subsquares in random Latin rectangles and extend the landscape of what is known for Latin squares and rectangles.

Abstract

Suppose that $k$ is a function of $n$ and $n\to\infty$. We show that with probability $1-O(1/n)$, a uniformly random $k\times n$ Latin rectangle contains no proper Latin subsquare of order $4$ or more, proving a conjecture of Divoux, Kelly, Kennedy and Sidhu. We also show that the expected number of subsquares of order 3 is bounded and find that the expected number of subsquares of order 2 is $\binom{k}{2}(1/2+o(1))$ for all $k\le n$.

Figures (2)

• Figure 1: An incomplete column cycle in a $4 \times 7$ Latin rectangle
• Figure 2: The Latin squares $M$ when $m\in\{4,5\}$. Colour coding $\;,,\;$ indicates cells to which Lemmas \ref{['l:newrcs']}, \ref{['l:colcycs']} and \ref{['l:rowcycs']}, respectively, will be applied in the proof of Theorem \ref{['t:subsq4boundgen']}.

Theorems & Definitions (29)

• Theorem 1.1
• Corollary 1.2
• Theorem 2.1
• Theorem 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4
• proof