Subsquares in Random Latin Rectangles

IF 1 2区数学 Q1 MATHEMATICS

Combinatorica Pub Date : 2025-05-14 DOI:10.1007/s00493-025-00156-0

Jack Allsop, Ian M. Wanless

引用次数: 0

Abstract

Suppose that k is a function of n and . 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 \(\left( {\begin{array}{c}k\\ 2\end{array}}\right) (1/2+o(1))\) for all \(k\leqslant n\).

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随机拉丁矩形中的子平方

假设k是n和的函数。我们以\(1-O(1/n)\)的概率证明了均匀随机\(k\times n\)拉丁矩形不包含4阶或4阶以上的适当拉丁子方，证明了Divoux， Kelly， Kennedy和Sidhu的一个猜想。我们还证明了3阶子平方的期望数目是有界的，并且发现对于所有\(k\leqslant n\)， 2阶子平方的期望数目是\(\left( {\begin{array}{c}k\\ 2\end{array}}\right) (1/2+o(1))\)。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Combinatorica 数学-数学

CiteScore

1.90

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： COMBINATORICA publishes research papers in English in a variety of areas of combinatorics and the theory of computing, with particular emphasis on general techniques and unifying principles. Typical but not exclusive topics covered by COMBINATORICA are - Combinatorial structures (graphs, hypergraphs, matroids, designs, permutation groups). - Combinatorial optimization. - Combinatorial aspects of geometry and number theory. - Algorithms in combinatorics and related fields. - Computational complexity theory. - Randomization and explicit construction in combinatorics and algorithms.