P(n，k)的计数及其良域

On the Calculation of P(n,k) and its Good Field

中文关键词:  全分拆  无序分拆  良城  计数.

英文关键词:total partitions  unordered partitions  good field  calculation.

中文摘要:

      设P(n，k)为整数n分为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)=P(n-k)；当k=1，4≤n≤5，或者当k≥2，2k+1≤n≤3k+2,P(n,k)=P(n-k)-(?)P(t)还定义了P(n，k)的良城，因面可借助若干个P(n)的值，迅速地计算大量的P(n，k)的值.

英文摘要:

      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. However, 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=1,4≤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) .