整边三角形与正整数的一类分拆数

The triangle with integer sides and a kind of partition number

正整数n的k部分分拆是将n表示成k个正整数的无序和.其中正整数n的3部分分拆的一个典型应用是整边三角形.对于整边三角形的研究已经有许多结果,对于周长为n的整边三角形个数有一个估计数公式T(n).本文作者利用分拆的Ferrers图将整边三角形与不定方程4x1+3x2+2x3=n联系起来,给出了利用T(n)计算正整数n的一类4部分分拆数的计数公式以及一类分部量不超过4的分拆数的计数公式,并讨论了其中一类分拆数在图论中的应用.

A partition of positive integer n is said to be the partition with k parts when it is representation of n as an unordered sum of k positive integers. The typical application of 3 parts of partition of n is the triangle with integer sides. There are many consequences of the triangle with integer sides. In particular, there is a simple calculating formula to calculate number of the triangle T（ n ） with integer sides which have perimeter n. In this paper, the anthors give a relation for the Diophantine equation 4x1 ＋ 3x2 ＋ 2x3 = n and the triangle with integer sides by Ferrers diagram of partition. With this relation, they give a calculating formula of a kind of partition with 4 parts by T（ n ） while a calculating formula of every parts is not exceed 4 is given. The application of a kind partition number in the graph theory is discussed also.