All-even Latin squares
type: Item
instance of: scholarly article
title: All-even Latin squares
zbMATH Open document ID: 0864.05017
DOI: 10.1016/S0012-365X(96)83015-9
published in: Discrete Mathematics
publication date: 1997-06-22T00:00:00Z
review text: Each row of a Latin square of order \(n\) or of a \(k\times n\) Latin rectangle with \(k\leq n\) can be thought of as a permutation of the elements in \(\{1,2,\dots,n\}\). The authors show that the proportion of Latin squares of order \(n\) for which all rows are even permutations is at most \(c^n\), where \(\sqrt{{3\over 4}}<c<1\). Coupled with work by Cameron and Kantor, this result establishes that for almost all Latin squares of order \(n\), the permutations given by the rows generate the symmetric group \(S_n\). The authors also show that for \(k\times n\) Latin rectangles with \(k\leq n-7\), the proportion of Latin rectangles for which all rows are even permutations is asymptotically equal to \(2^{-k}\), thereby indicating a ``quasi-random'' behavior for the rows.
Mathematics Subject Classification ID: 05B15
zbMATH DE Number: 935380
zbMATH Keywords: Latin square
zbMATH Keywords: Latin rectangle
zbMATH Keywords: permutation
zbMATH Keywords: even permutations
zbMATH Keywords: symmetric group
author: Roland Häggkvist
author: Jeannette Janssen
reviewed by: Chester J. Salwach
MaRDI profile type: Publication
cites work: Almost all quasigroups have rank 2
cites work: Research problems
cites work: Q4347884
cites work: A lower bound for the permanent of a doubly stochastic matrix
cites work: On even and odd latin squares
full work available at URL: s0012-365x(96)83015-9
OpenAlex ID: W2093141448
description: scientific article; zbMATH DE number 935380