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| Title: | 一個卡特蘭等式的組合證明
A Combinatorial Proof of a Catalan Identity |
| Authors: | 劉映君 |
| Contributors: | 李陽明
劉映君 |
| Keywords: | 卡特蘭等式
Catalan Identity |
| Date: | 2017 |
| Issue Date: | 2017-08-10 09:57:57 (UTC+8) |
| Abstract: | 在這篇論文裡，我們探討卡塔蘭等式 (n + 2)Cn+1 = (4n + 2)C2 的證明
方法。以往都是用計算的方式來呈現卡塔蘭等式的由來，但我們選擇用組合
的方法來證明卡塔蘭等式。
這篇論文主要是應用 Cn+1 壞路徑對應到打點 Cn 好路徑以及 Cn+1 好路
徑對應到打點 Cn 壞路徑的⽅式來證明卡特蘭等式。
In this thesis, we give another approach to prove Catalan identity,
(n + 2)Cn+1 = (4n + 2)C2. In the past we use the method of computation to show Catalan Identity, in this thesis we choose a combinatorial proof of the Catalan identity.
This thesis is primary using the functions from Cn+1 totally bad path to Cn dotted good path, and from Cn+1 good path to Cn dotted totally bad path. |
| Reference: | Ronald Alter. Some remarks and results on Catalan numbers. pages 109–132, 1971.
[2] Ronald Alter and K. K. Kubota. Prime and prime power divisibility of Catalan numbers.
J. Combinatorial Theory Ser. A, 15:243–256, 1973.
[3] Federico Ardila. Catalan numbers. Math. Intelligencer, 38(2):4–5, 2016.
[4] Young-Ming Chen. The Chung-Feller theorem revisited. Discrete Math., 308(7):1328–
1329, 2008.
[5] Ömer E ̆gecioğlu. A Catalan-Hankel determinant evaluation. In Proceedings of the Fortieth
Southeastern International Conference on Combinatorics, Graph Theory and Computing,
volume 195, pages 49–63, 2009.
[6] R. Johnsonbaugh. Discrete Mathematics. Pearson/Prentice Hall, 2009.
[7] Thomas Koshy. Catalan numbers with applications. Oxford University Press, Oxford,
2009.
[8] Tamás Lengyel. On divisibility properties of some differences of the central binomial
coefficients and Catalan numbers. Integers, 13:Paper No. A10, 20, 2013.
[9] Youngja Park and Sangwook Kim. Chung-Feller property of Schröder objects. Electron.
J. Combin., 23(2):Paper 2.34, 14, 2016.
[10] Matej ̌Crepin ̌sek and Luka Mernik. An efficient representation for solving Catalan number
related 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
103751014 |
| Source URI: | http://thesis.lib.nccu.edu.tw/record/#G0103751014 |
| Data Type: | thesis |
| Appears in Collections: | [應用數學系] 學位論文
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| 101401.pdf | 1052Kb | Adobe PDF2 | 483 | View/Open |
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