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Title: 一個卡特蘭等式的重新審視
A Catalan Identity revisited
Authors: 李珮瑄
Contributors: 李陽明
Chen, Young-Ming
Keywords: 卡特蘭等式
Dyck 路徑
Catalan identity
Dyck path
Date: 2020
Issue Date: 2020-08-03 17:57:15 (UTC+8)
Abstract: 本篇論文探討卡特蘭等式(n+2)Cn+1=(4n+2)Cn 證明方式以往都以計算方式推導得出,當我參加劉映君的口試時,發現她使用組合方法來證明這個等式。當我在尋找論文的主題時,讀到李陽明老師的一篇論文"The Chung Feller theorem revisited",發現Dyck 路徑也可以作為卡特蘭等式的組合證明,因此我們完成(n+2)Cn+1=(4n+2)Cn 的組合證明。
通過Dyck 路徑證明卡特蘭等式可以得到以下優勢:
由於x2 中的Q 的子路徑A為空,因此在Bu交換和Ad部分後,生成新的提
3.在計算函數g1(g2) 的反函數的過程中,缺陷(提升)恢復模式必遵循
When we first prove the Catalan identity, (n+2)Cn+1=(4n+2)Cn. We often prove it by calculation. When I participated in the oral examination of Ying-Jun Liu’s essay, I found that she used a combinatorial proof to prove this identity.When I was looking for the subject of the thesis, I read a paper by professor Young-Ming Chen, "The Chung Feller theorem revisited", which found that Dyck paths could also be used as a combinatorial proof of the Catalan identity. Therefore, we completed the combinatorial proof of (n+2)Cn+1=(4n + 2)Cn.
Proving the Catalan identity through the Dick paths can reveal the following advantages:
1.The subpath C does not change during the process of
switching of the portions Ad and Bu.
2.Since the subpath B of P in x1 is empty, a new flaw
generated after switching of the portions Ad and Bu must
be followed by the original subpath C.
Since the subpath A of Q in x2 is empty, a new lift
generated after switching of the portions Bu and Ad must
be followed by the original subpath C.
3.In the process of computing the preimage of a function g1
(g2), the flaws (lifts) recovery mode follows the "Last in First out" or "First in Last out".
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Description: 碩士
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Data Type: thesis
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