Please use this identifier to cite or link to this item: https://ah.nccu.edu.tw/handle/140.119/131112


Title: 關於一個組合等式的對射證明
A Bijection Proof About a Combinatorial Identity
Authors: 黃大維
HUANG, TA-WEI
Contributors: 李陽明
Chen, Young-Ming
黃大維
HUANG, TA-WEI
Keywords: 組合等式
對射證明
排列組合
Combinatorial identity
Bijection proof
Combinatorics
Date: 2020
Issue Date: 2020-08-03 17:58:35 (UTC+8)
Abstract: 本篇論文的主旨是要證明兩個排列組合n*C(2r,(n+r-1))*C(r,2r)與C(r,n)*(n-r)*C(r,(n+r-1))是否相
等,並嘗試找出這兩個排列組合之間的關係與意義。
本篇論文提供兩個證明此組合等式n*C(2r,(n+r-1))*C(r,2r)=C(r,n)*(n-r)*C(r,(n+r-1))的方法,第一個方法簡潔快速,但是只能單純證明等式的相等,無法看出左式與右式的關係。
第二個方法是將左式與右式分別建構成一個集合,並在兩個集合之間建構
一個函數。此函數的特性為一對一(one to one)且映成(onto),也就是對射函數(bijection function),利用此方法完成第二個方法的證明。
The main topic in this paper is to prove the equality of these two expressions n*C(2r,(n+r-1))*C(r,2r)and C(r,n)*(n-r)*C(r,(n+r-1)).And attempt to find the relation between these two expressions and their meaning in combinatorics.
In this paper, we provide two kinds of methods to solve the combinatorial identity n*C(2r,(n+r-1))*C(r,2r)=C(r,n)*(n-r)*C(r,(n+r-1)).The first way is simply proving the combinatorial identity by expansing the formula without knowing the meaning of the equation.
The second method to solve the equation is constructing two sets consisting of the numbers of elements in n*C(2r,(n+r-1))*C(r,2r) and C(r,n)*(n-r)*C(r,(n+r-1)), respectively.And construct a bijective function to complete the second method of proof.
Reference: [1] Alan Tucker(2012),Applied Combinatorics,sixth edition,John Wiley & Sons,Inc.,p.233.
[2]C.L.Liu(2000),Introduction to Combinatorial Mathematics(International editions),McGraw-Hill.
[3]Susanna S.Epp(2003),Discrete Mathematics with Applications,Cengage Learning
[4]Peter J.Cameron(1994),Combinations:Topics,Techniques,Algorithms,London,
Queen Mary&Westfield College.
[5]劉麗珍(1994),一個組合等式的一對一證明,國立政治大學,應用數學碩士論文。
[6]陳建霖(1996),一個組合等式的證明,國立政治大學,應用數學碩士論文。
[7]韓淑惠(2011),開票一路領先的對射證明,國立政治大學,應用數學碩士論文。
[8]薛麗姿(2013),一個珠狀排列的公式,國立政治大學,應用數學碩士論文。
[9]黃永昌(2018),一個組合等式的對射證明,國立政治大學,應用數學碩士論文。
Description: 碩士
國立政治大學
應用數學系
108751008
Source URI: http://thesis.lib.nccu.edu.tw/record/#G0108751008
Data Type: thesis
Appears in Collections:[應用數學系] 學位論文

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