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研究生： 汪哲宇
Wang, Che-Yu
論文名稱： Catalan 族的對射關係及其應用
Bijections and Applications For Catalan Family
指導教授： 黃柏嶧
Huang, Po-Yi
學位類別： 碩士
Master
系所名稱： 理學院 - 數學系應用數學碩博士班
Department of Mathematics
論文出版年： 2013
畢業學年度： 101
語文別： 英文
論文頁數： 46
中文關鍵詞： 對射關係 、Catalan 數 、Fuss-Catalan 數
外文關鍵詞： Bijection, Catalan Numbers, Fuss-Catalan Numbers
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    • 本文藉由探討一些關於已知Catalan 數的計數問題，尋找各問題間的對射關係，並且探討Catalan 數的推廣- Fuss-Catalan 數，如：在Catalan 數問題中的312-avoiding推廣到Fuss-Catalan 數的問題。

    We discuss some known counting problems for Catalan numbers in this thesis. First, we find the bijection relations between each problem. Then we find generalized result for some problems known as Fuss-Catalan problems. For instance, numbers the 312-avoiding problem for Catalan numbers can be generalized for Fuss-Catalan numbers.

    1 Introduction 8 2 Counting Problems for Catalan Numbers 10 2.1 Dyck Paths 10 2.2 Balanced Parentheses 12 2.3 Standard Young Tableaux 12 2.4 Noncrossing Matchings 13 2.5 Noncrossing Chords 13 2.6 312-avoiding 14 2.7 Multisets 15 2.8 Plane Binary Trees 16 2.9 Tiling of The Staircase Shape 16 2.10 Non-positive Partial Sum 17 2.11 Pairs of Lattice paths 18 3 Bijections 19 3.1 Balanced Parentheses, Noncrossing Matchings and Multisets 19 3.2 Balanced Parentheses, Noncrossing Matching and Standard Young Tableaux 19 3.3 Noncrossing Chords and Plane Binary Trees 20 3.4 Noncrossing Chords, Balanced Parentheses, Noncrossing Matchings and Multisets 21 3.5 Dyck Paths, Balanced Parentheses, Multisets and Standard Young Tableaux 22 3.6 Balanced Parentheses, Noncrossing Matching, Plane Binary Trees and Tiling of The Staircase Shape 24 3.7 Dyck Paths, Multisets and 312-avoiding 25 3.8 Dyck Paths and Non-positive Partial Sum 26 3.9 Pairs of Lattice paths and Balanced Parentheses 27 4 Generating Functions 29 5 Generalizations 36 5.1 Extension of Problems 36 5.2 Bijections 40 5.2.1 Plane Ternary Trees and Noncrossing Triangles 41 5.2.2 Noncrossing Triangles, 3-Multiset and 3-Noncrossing Matching 41 5.2.3 2-Dyck Path, 3-Multiset and 312-avoiding on 2-Multiset 43 5.2.4 2-Dyck Path and Non-Positive Partial Sums 43 References 45

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