一类分拆数的计算问题

Unit Root Test of AR (1) Model

正整数n的分拆是指将正整数n表示成一个或多个正整数的无序和.设Q(n,m)是将正整数n分拆为m个互不相同的正整数之和的无序分拆数,而P(n,m)是将正整数n分拆成m个部分的无序分拆的分拆数.它们都是组合,图论,数论的重要概念和数据.本文得到了关于Q(n,m)的一个递推关系以及P(n,m)与Q(n,m)之间的直接关系,进而可以利用已有的一些结果来计算Q(n,m)的值.同时本文也讨论了Q(n,m)在图论中的一个应用.

A partition of positive integer n is representation of n as unordered sum of one or more positive integers. Let Q（n, m） be the number of unordered partitions of an integer n into m distinct positive integers. And let P （n, m） be the number of unordered partitions of a positive integer n into m parts. They are all the important concepts in Combinatorics, Graph theory and Number theory. In this paper, a recurrence relation that Q（n, m） satisfies is given. The relation between P（n ,m） and Q（n ,m） is got. Thus we can compute the value of Q（n ,m） by the conclusions which have been got. And we also study the application of Q（n ,m） in Graph theory.