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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LEE, PEI-SHIUAN Contributors: 李陽明Chen, Young-Ming李珮瑄LEE,PEI-SHIUAN Keywords: 卡特蘭等式Dyck 路徑Catalan identityDyck 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：1.子路徑C在切換過程中不會改變。2.由於x1中的P的子路徑B為空，因此在交換Ad和Bu部分後，生成新的缺陷必連接在原始子路徑C之後。由於x2 中的Q 的子路徑A為空，因此在Bu交換和Ad部分後，生成新的提升必連接在原始子路徑C之後。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"後進先出"或"先進後出"規則。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 ofswitching of the portions Ad and Bu.2.Since the subpath B of P in x1 is empty, a new flawgenerated after switching of the portions Ad and Bu mustbe followed by the original subpath C.Since the subpath A of Q in x2 is empty, a new liftgenerated after switching of the portions Bu and Ad mustbe 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". Reference: [1] 劉映君. 一個卡特蘭等式的組合證明, 2017.[2] Ronald Alter. Some remarks and results on catalan numbers. 05 2019.[3] Ronald Alter and K.K Kubota. Prime and prime power divisibility of catalan numbers.Journal of Combinatorial Theory, Series A, 15(3):243 – 256, 1973.[4] Federico Ardila. Catalan numbers. The Mathematical Intelligencer, 38(2):4–5, Jun 2016.[5] Young-Ming Chen. The chung–feller theorem revisited. Discrete Mathematics, 308:1328–1329, 04 2008.[6] Ömer Eğecioğlu. A Catalan-Hankel determinant evaluation. In Proceedings of the FortiethSoutheastern International Conference on Combinatorics, Graph Theory and Computing,volume 195, pages 49–63, 2009.[7] R. Johnsonbaugh. Discrete Mathematics. Pearson/Prentice Hall, 2009.[8] Thomas Koshy. Catalan numbers with applications. Oxford University Press, Oxford,2009.[9] Tamás Lengyel. On divisibility properties of some differences of the central binomialcoefficients and Catalan numbers. Integers, 13:Paper No. A10, 20, 2013.[10] Youngja Park and Sangwook Kim. Chung-Feller property of Schröder objects. Electron.J. Combin., 23(2):Paper 2.34, 14, 2016.[11] Matej Črepinšek and Luka Mernik. An efficient representation for solving Catalan numberrelated problems. Int. J. Pure Appl. Math., 56(4):589–604, 2009. Description: 碩士國立政治大學應用數學系104751012 Source URI: http://thesis.lib.nccu.edu.tw/record/#G0104751012 Data Type: thesis Appears in Collections: [應用數學系] 學位論文

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