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 identityBijection proofCombinatorics 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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