On Even and Odd Latin Squares
On Even and Odd Latin Squares
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关于偶数和奇数拉丁方
DOI：
10.1016/0097-3165(95)90115-9
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发表时间：
1995
期刊：
影响因子：
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通讯作者：
J. Janssen
中科院分区：
文献类型：
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作者：
J. Janssen
根据其行和列所给出的排列符号，拉丁正方形可以被分类为奇数或偶数。本文分析了拉丁方在同位素群（行、列和符号调换）和变换群（行、列和符号调换）作用下的奇对性。本文给出了变换群作用下的偶对的行为完全确定的规则，并且只依赖于n阶。同时，我们简短地证明了Alon-Tarsi猜想和Huang-White猜想的等价性，并给出了计算机计算的结果，证明了8阶拉丁偶数和奇数的平方不相等，从而证明了这个特殊阶的Alon-Tarsi猜想。
Latin squares can be classified as odd or even according to the signs of the permutations given by their rows and columns. In this paper, the behaviour of the parities of a latin square under the action of the isotopy group (permuting rows, columns, and symbols) and the transformation group (interchanging rows, columns, and symbols) is analyzed. A rule is given that shows that the behaviour of parities under the action of the transformation group is completely determined, and only depends on the order n. Also, we give a short proof of the equivalence of the conjectures of Alon-Tarsi and Huang-White and present the results of a computer calculation that show that the number of even and odd latin squares of order 8 are not equal, thus proving the Alon-Tarsi conjecture for this particular order.