Concerning a certain set of arrangements

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.