Please use this identifier to cite or link to this item:
https://ah.lib.nccu.edu.tw/handle/140.119/131105
DC Field | Value | Language |
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dc.contributor.advisor | 李陽明 | zh_TW |
dc.contributor.advisor | Chen, Young-Ming | en_US |
dc.contributor.author | 李珮瑄 | zh_TW |
dc.contributor.author | LEE,PEI-SHIUAN | en_US |
dc.creator | 李珮瑄 | zh_TW |
dc.creator | LEE, PEI-SHIUAN | en_US |
dc.date | 2020 | en_US |
dc.date.accessioned | 2020-08-03T09:57:15Z | - |
dc.date.available | 2020-08-03T09:57:15Z | - |
dc.date.issued | 2020-08-03T09:57:15Z | - |
dc.identifier | G0104751012 | en_US |
dc.identifier.uri | http://nccur.lib.nccu.edu.tw/handle/140.119/131105 | - |
dc.description | 碩士 | zh_TW |
dc.description | 國立政治大學 | zh_TW |
dc.description | 應用數學系 | zh_TW |
dc.description | 104751012 | zh_TW |
dc.description.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。\n通過Dyck 路徑證明卡特蘭等式可以得到以下優勢:\n1.子路徑C在切換過程中不會改變。\n2.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\n必連接在原始子路徑C之後。\n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\n升必連接在原始子路徑C之後。\n3.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\n"後進先出"或"先進後出"規則。 | zh_TW |
dc.description.abstract | 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.\nProving the Catalan identity through the Dick paths can reveal the following advantages:\n1.The subpath C does not change during the process of\nswitching of the portions Ad and Bu.\n2.Since the subpath B of P in x1 is empty, a new flaw\ngenerated after switching of the portions Ad and Bu must\nbe followed by the original subpath C.\nSince the subpath A of Q in x2 is empty, a new lift\ngenerated after switching of the portions Bu and Ad must\nbe followed by the original subpath C.\n3.In the process of computing the preimage of a function g1\n(g2), the flaws (lifts) recovery mode follows the "Last in First out" or "First in Last out". | en_US |
dc.description.tableofcontents | Contents\n致謝 ii\n中文摘要 iii\nAbstract iv\nContents v\nList of Figures vi\n1 Introduction 1\n2 Paths Start with Up-step 3\n3 Paths Start with Down-step 14\n4 Summary 25\nAppendix A examples of Catalan identity 26\nA.1 (n+2)Cn+1=(4n+2)Cn 26\nBibliography 30 | zh_TW |
dc.format.extent | 501785 bytes | - |
dc.format.mimetype | application/pdf | - |
dc.source.uri | http://thesis.lib.nccu.edu.tw/record/#G0104751012 | en_US |
dc.subject | 卡特蘭等式 | zh_TW |
dc.subject | Dyck 路徑 | zh_TW |
dc.subject | Catalan identity | en_US |
dc.subject | Dyck path | en_US |
dc.title | 一個卡特蘭等式的重新審視 | zh_TW |
dc.title | A Catalan Identity revisited | en_US |
dc.type | thesis | en_US |
dc.relation.reference | [1] 劉映君. 一個卡特蘭等式的組合證明, 2017.\n[2] Ronald Alter. Some remarks and results on catalan numbers. 05 2019.\n[3] Ronald Alter and K.K Kubota. Prime and prime power divisibility of catalan numbers.\nJournal of Combinatorial Theory, Series A, 15(3):243 – 256, 1973.\n[4] Federico Ardila. Catalan numbers. The Mathematical Intelligencer, 38(2):4–5, Jun 2016.\n[5] Young-Ming Chen. The chung–feller theorem revisited. Discrete Mathematics, 308:1328–\n1329, 04 2008.\n[6] Ömer Eğecioğlu. A Catalan-Hankel determinant evaluation. In Proceedings of the Fortieth\nSoutheastern International Conference on Combinatorics, Graph Theory and Computing,\nvolume 195, pages 49–63, 2009.\n[7] R. Johnsonbaugh. Discrete Mathematics. Pearson/Prentice Hall, 2009.\n[8] Thomas Koshy. Catalan numbers with applications. Oxford University Press, Oxford,\n2009.\n[9] Tamás Lengyel. On divisibility properties of some differences of the central binomial\ncoefficients and Catalan numbers. Integers, 13:Paper No. A10, 20, 2013.\n[10] Youngja Park and Sangwook Kim. Chung-Feller property of Schröder objects. Electron.\nJ. Combin., 23(2):Paper 2.34, 14, 2016.\n[11] Matej Črepinšek and Luka Mernik. An efficient representation for solving Catalan number\nrelated problems. Int. J. Pure Appl. Math., 56(4):589–604, 2009. | zh_TW |
dc.identifier.doi | 10.6814/NCCU202000719 | en_US |
item.openairecristype | http://purl.org/coar/resource_type/c_46ec | - |
item.grantfulltext | open | - |
item.fulltext | With Fulltext | - |
item.openairetype | thesis | - |
item.cerifentitytype | Publications | - |
Appears in Collections: | 學位論文 |
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