A Combinatorial Problem
A Combinatorial Problem
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组合问题
DOI：
10.1002/sapm1933121321
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发表时间：
1933
期刊：
影响因子：
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通讯作者：
J. Todd
中科院分区：
文献类型：
--
作者：
J. Todd
这个问题相当于找到一个4 n行和列的正交矩阵，其中每个元素都是+1或-1。考虑这样一个矩阵。通过改变一定数量的行和列的元素的符号，它可以简化为标准形式，其中第一行和第一列中的每个元素都是+1。那么正交性条件要求在通过抑制第一行和第一列而获得的4 n-1行和第一列的矩阵中，每行应包含2n-1个元素+1，并且任何两行应仅具有出现在同一列中的n-1对元素+1。相反，很明显，从4 n-1个物体以上述方式的每一种排列，我们都可以推导出一个正交方阵。在前面的论文中，佩利先生给出了当4 n是属于某些特殊类型之一时构造这样一个矩阵的一般规则。本文详细讨论了较简单的情况，并顺便表明，这些一般方法绝不是穷尽的可能性。在这些简单的情况下，我们还将考虑（对象的）排列群，在这些排列群下，集合中的排列保持不变。
The problem is equivalent to finding an orthogonal matrix of 4n rows and columns every element of which is either+ 1 or-1. For consider such a matrix. By changing the signs of the elements of a certain number of rows and columns it may be reduced to a standard form in which every element in the first row and column is+ 1. Then the condition of orthogonality requires that in the matrix of 4n-1 rows and columns obtained by suppressing the first row and column, each row should contain 2n-1 elements+ 1, and any two rows should have just n-1 pairs of elements+ 1 which occur in the same column. Conversely, it is clear that from every arrangement of 4n-1 objects in sets in the manner described we can deduce a square orthogonal matrix. In the preceding paperl Mr. Paley has given general rules for constructing such a matrix when 4n is a number belonging to one of certain special types. The present paper deals in detail with the simpler cases, and shows incidentally that these general methods by no means exhaust the possibilities. We shall also consider, in these simple cases, the group of permutations (of the objects) under which the arrangement in sets remains invariant.