快速造P(n,k)大表的左肩法则和斜线法则

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，它为大师欧拉所建立（１７０７－１７８３）．它是组合图论和数论里最重要的数据之一．然而，它却十分难于计数和造表．本文，由公式Ｐ（ｎ，ｋ）＝Ｐ（ｎ－１，ｋ－１）＋Ｐ（ｎ－ｋ，ｋ）定义了Ｐ（ｎ，ｋ）的左肩数和锐角数，并由此得到求Ｐ（ｎ，ｋ）的左肩法则（第一法则）．还根据本文作者［５］的一些重要定理得到求 Ｐ（ｎ，ｋ）的斜线法则（第二法则）．使用这些法则得到造Ｐ（ｎ，ｋ）大表的有趣原理．为方便计，我们仅用第一法则设计了计算机程序，用此程序即可快速造出任意大的Ｐ（ｎ，ｋ）表．

On the Law of Left Shoulder and the Law of Oblique Line for Constructing a Large Table of P(n, k) Quickly

Let P(n, k) be the number of unordered partitions of an integer n into k parts where each part ≥1, set by master Euler (1707-1783). It is one of most important numbers in Combinatorics, Graph theory and Number theory.However, it is rather difficult to find the values of P(n, k) and to construct a large table of P(n, k). In this paper, from this formula P(n, k) = P(n - 1, k - 1) + P(n - k, k) we define the "number of left shoulder" of P(n, k) and the "acute number" of P(n, k), by which we get at the "law of left shoulder" to find P(n, k) (First law), also the "law of oblique line" to find P(n, k) (Second law) on the basis of some important theorems of the paper Wu Qiqi 5]. By using these laws we obtaine an ioteresting principle for constructing a large table of P(n, k). For the sake of convenience, we use the only first law to design the proceeding of calculator, by which we can quickly construct an arbitrary large table of P(n, k).