Please use this identifier to cite or link to this item: https://ah.nccu.edu.tw/handle/140.119/118828


Title: 一個組合等式的對射證明
A Bijective Proof of a Combinatorial Identity
Authors: 黃永昌
Huang, Young-Chang
Contributors: 李陽明
Chen, Yong-Ming
黃永昌
Huang, Young-Chang
Keywords: 對射
組合等式
Date: 2018
Issue Date: 2018-07-24 11:00:52 (UTC+8)
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:
C(r,n)*(n-r)*C(r,(n+r-1))=n*C(2r,(n+r-1))*C(r,2r)

  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:

C(r,n)*(n-r)*C(r,(n+r-1))=n*C(2r,(n+r-1))*C(r,2r)

  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.
Reference: [1] Alan Tucker, Applied Combinatorics,sixth edition, John Wiley & Sons,Inc.,p.233,2012.
[2] 劉麗珍,一個組合等式的一對一證明,政治大學應用數學碩士論文,1994。
[3] 陳建霖,一個組合等式的證明,政治大學應用數學碩士論文,1996。
[4] 韓淑惠,開票一路領先的對射證明,政治大學應用數學碩士論文,2011。
[5] 薛麗姿,一個珠狀排列的公式,政治大學應用數學碩士論文,2013。
Description: 碩士
國立政治大學
應用數學系
104751004
Source URI: http://thesis.lib.nccu.edu.tw/record/#G0104751004
Data Type: thesis
Appears in Collections:[應用數學系] 學位論文

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