- 學位論文
禁止模式的默慈金路徑與其統計量之聯合分布
Pattern-avoiding Motzkin paths and their joint distributions of statistics
摘要
從Asinowski, Banderier, Roitner等人撰寫的論文中，他們研究了具有禁止模式Motzkin path。不同禁止模式的Motzkin path具有類似的計數數列，他們研究的方向是分別計算出不同禁止模式Motzkin path的生成函數。而淡江大學研究生陳麒文，也經徐祥峻教授指導，對其禁止模式有了新的進展，透過研究生陳麒文的論文，發現那些相同的數列卻禁止不同模式的Motzkin path 存在著一一對應的關係，像是禁止UU模式n步的Motzkin path 對應禁止UD模式n+1步的Motzkin path ，以統計量等的方式可以知道DHU會與UHU等分布，也能使用遞迴的方式尋找如何產生一一對應。 本篇主要探討禁止DU模式n步下的Mitzkin paths 與禁止UD模式n+1步下的Mitzkin path、禁止DD,DU模式n步的Motzkin path對應禁止DD,UD模式n+1步的Motzkin path，以及禁止HD,DU模式n步的Motzkin paths 對應禁止DH,UD模式n+1步的Motzkin path 均存在一一對應，以及使用多組統計量的方式來尋找等分布，最後運用鏡射的方式來解決更多的一一對應。
並列摘要
From the paper by Asinowski, Banderier, Roitner et al., they study the Motzkin path with pattern-avoiding. Motzkin paths with different pattern-avoiding have similar counting sequences. The direction of their research is to calculate the generating functions of different pattern-avoiding Motzkin paths, and Tamkang University graduate student Chi-Wen, Chen, under the guidance of Professor Hsiang-Chun, Hsu, has made new progress in the pattern-avoiding. Through the thesis of graduate student Chi-Wen, Chen, it is found that there is a one-to-one correspondence between the Pattern-avoiding Motzkin paths of the same sequences, such as n-steps UU-avoiding Motzkin paths and (n+1)-steps UD-avoiding Motzkin paths, we can know statistic DHU and UHU are equal distributions, and also use recursive methods to find how to generate one-to-one correspondence. This paper mainly discusses n-steps DU-avoiding Motzkin paths and (n+1)-steps UD-avoiding Motzkin paths, and n-steps DD,DU-avoiding Motzkin paths and (n+1)-steps DD,UD-avoiding Motzkin paths, and n-steps DH,DU-avoiding Motzkin paths and (n+1)-steps HD,UD-avoiding Motzkin paths. And used their joint distributions of statistics to find equal distributions, finally use mirroring to solve more statistic is equal joint distributions and how to one-to-one.