一個卡特蘭等式的重新審視

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。

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".