摘要
设Ｐ（ｎ，ｋ）为整数ｎ分ｋ部的无序分拆的个数，每个分部≥１；Ｐ（ｎ）为ｎ的全分拆的个数．Ｐ（ｎ，ｋ）是用途广泛的、且又十分难予计算的数．本文证明了下述定理：当ｎ＜ｋ，Ｐ（ｎ，ｋ）＝０；当ｋ≤ｎ≤２ｋ，Ｐ（ｎ－ｋ）；当 ｋ＝１，４≤ｎ≤５，或者当ｋ≥２，２ｋ＋１≤ｎ≤３ｋ＋２，Ｐ（ｎ，ｋ）－Ｐ（ｔ）。还定义了Ｐ（ｎ，ｋ）的良域，因而可借助若干个Ｐ（ｎ）的值，迅速地计算大量的Ｐ（ｎ，ｋ）的值。
Let P(n,k) be the number of unordered partitions of an integer n into k parts,where each part≥ 1, and Ｐ(n) the number of all unordered partitions of n (so,in brief, it is called the number of total partitions). The number Ｐ(n,k) has a broad applications. Howev-er, it is rather difficult to find the values of P(n,k).In this paper we give the following theorem: p(n,k) = 0, when n < k ; p(n,k)=P(n- k),when k≤n ≤2k;and p(n,k) P(n - k) P(t),when k= l, 4≤ n≤5,or when k≥2,2k + 1≤ n≤ 3k + 2. And we define also the good field of p(n,k). This theorem will help us find numberless the values of P(n,k) quickly with the aid of the values of P(n).
