Double Counting 19 计拔 浦亮

1 Double Counting 什么是Double Couting? 2 以Counting Trees �例 3 关于Counting Trees 的一些小拓展

什么是Double Couting? In combinatorics, double counting, also called counting in two ways, is a combinatorial proof technique for showing that two expressions are equal by demonstrating that they are two ways of counting the size of one set. In this technique, which van Lint & Wilson (2001) call “one of the most important tools in combinatorics, ” one describes a finite set X from two perspectives leading to two distinct expressions for the size of the set. Since both expressions equal the size of the same set, they equal each other. Double Counting，通俗讲即算两次，通过对同一集合进行两次不同形式 的表述，令两个表述相等，从而可以得出一些原本难以得到的表达式，通常 用于组合数学中。

以Counting Trees 为例 在集合N={1，2，3，……n}这个顶点集上可以形成多少颗不同的标记树？ Cayley's formula implies that there is 1 = 22 − 2 tree on two vertices, 3 = 33 − 2 trees on three vertices, 16 = 44 − 2 trees on four vertices 是否可以猜出Tn的表达式？

参考资料： 维基百科： https: //en. wikipedia. org/wiki/Double_counting_(proof_technique)#Counting_trees 《数学天书中的证明》第四版

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