一個組合等式的對射證明

Abstract

研究組合數學的目的不僅是算出答案，而是要理解算出答案的過程。在本篇論文中，本研究嘗試用組合的方法證明以下等式：\nC(r,n)*(n-r)*C(r,(n+r-1))=n*C(2r,(n+r-1))*C(r,2r)\n\n 在解這個組合等式的時候，我們不使用一般的展開計算方式，而是先建構兩個集合，其個數分別為 C(r,n)*(n-r)*C(r,(n+r-1)) 以及 n*C(2r,(n+r-1))*C(r,2r) ，並在兩個集合之間建構一個函數。此函數的特點是一對一且映成，也就是說此函數為對射函數(bijective function)，利用這個方法即可完成本篇的證明。

The purpose of studying combinatory mathematics is not only to calculate the answer, but to understand the process of calculating the answer. In this paper, this study attempts to use the combined method to prove the following equation:\n\nC(r,n)*(n-r)*C(r,(n+r-1))=n*C(2r,(n+r-1))*C(r,2r)\n\n To solve this combination equation, instead of using the general expansion calculation method, two sets are constructed whose numbers of elements are, respectively,C(r,n)*(n-r)*C(r,(n+r-1)) and n*C(2r,(n+r-1))*C(r,2r), then a function are constructed between two sets. This function is characterized by a one to one and onto, that is to say this function is a bijective function. We can use this method to complete the proof of this article.