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.

[1] 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.