Concerning a certain set of arrangements
Concerning a certain set of arrangements
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DOI：
10.1090/s0002-9939-1950-0038922-4
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发表时间：
1950-06
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影响因子：
--
通讯作者：
Ben Dushnik
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文献类型：
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作者：
Ben Dushnik
在S中存在一种安排，其中ak跟随所有的a，其中i < k。这样的集合S必定存在;例如，其终端元素分别为1，2，.，m的任何一组m个排列显然将是k-适合于任何k？M.最小的基数N使得存在一个N个排列的集合，该集合是k适合于m的，将由N（m，k）表示;因此，对于任何m和k，这是一个定义良好的正整数，k <m。在前m个正整数的排列的任何集合中，几个排列的终端元素的集合将被称为端元素，而所有其他元素则被称为中间元素。如果上下文清楚地指出了k的值，或者如果引用适用于所有的k值，则我们将在下文中引用“合适的”布置集合（省略数字前缀）。在本文中，我们主要关注的是N（m，k）的评估。评价是不完全的;主要结果在定理I中给出。在最后一节中，我们使用所获得的结果来制定一个问题的答案与偏序集。在本文中，M将表示前m个正整数的集合。小写字母，只要它们代表数字，就代表正整数。
there exists an arrangement in S in which ak follows all the a, with i < k. Such sets S surely exist; for example, any set of m arrangements whose terminal elements are 1, 2, , m, respectively, will obviously be k-suitable for any k ? m. The smallest cardinal number N such that there exists a set of N arrangements which is k-suitable for m will be denoted by N(m, k); this is therefore a well defined positive integer for any m and k, k <m. In any collection of arrangements of the first m positive integers, the set of the terminal elements of the several arrangements will be called the end-elements, while all the other elements will then be referred to as mid-elements. We shall hereafter refer to 'suitable" sets of arrangements (omitting the numerical prefix) if the value of k is clearly indicated by the context, or if the reference applies to all values of k. In this paper we are primarily concerned with the evaluation of N(m, k). The evaluation is not complete; the main result is given in Theorem I. In the last sections we use the results obtained to formulate an answer to a problem in connection with partially ordered sets. Throughout this paper, M will denote the set of the first m positive integers. Small letters, insofar as they represent numbers, will represent positive integers.