National Taiwan Normal University Course Outline
 Spring , 2022

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I.Course information
Serial No. 2719 Course Level Undergraduate / Master
Course Code MAC9025 Chinese Course Name 組合數學（IB）
Course Name Combinatorics Mathematics (IB)
Department Department of Mathematics
Two/one semester 1 Req. / Sel. Sel.
Credits 3.0 Lecturing hours Lecture hours: 3
Prerequisite Course ◎1. This is a cross-level course and is available for junior and senior undergraduate students, master's students and PhD students. 2. If the listed course is a doctroal level course, it is only available for master's students and PhD students.
Comment
Course Description
Time / Location Mon. 2-4 Gongguan MA3-10
Curriculum Goals Corresponding to the Departmental Core Goal
1.　Solving applications by using counting skills. College:
　1-1 Equipped with professional mathematics competences
　1-2 Being able to reason and induct with mathematical logic
　1-5 Being able to use mathematics as tools to learn other subjects
Master:
　1-1 Equipped with professional mathematics competences
　1-2 Being able to reason and induct with mathematical logic
　1-5 Being able to use mathematics as tools to learn other subjects
2.　Using combinatorics to solve problems in related fields. College:
　1-5 Being able to use mathematics as tools to learn other subjects
Master:
　1-5 Being able to use mathematics as tools to learn other subjects
3.　Transferring between models in combinatorics. College:
　1-4 Possessing the abilities to propose and solve questions in advanced mathematics
　3-1 Being able to seek out answers with the attitudes of patience, diligence, concentration, and curiosity
Master:
　1-4 Possessing the abilities to propose and solve questions in advanced mathematics
　3-1 Being able to seek out answers with the attitudes of patience, diligence, concentration, and curiosity

II. General Syllabus
Instructor(s) LIN, Yen-Chi/ 林延輯
Schedule
  1. 基礎計數原則；加法原理，乘法原理，一一對應原理 (1 週)
  2. 二項式係數 (2週)
  3. 生成函數與遞迴關係 (1 週)
  4. 集合的分割與第二類 Stirling 數 (1 週)
  5. 排列與第一類 Stirling 數 (1 週)
  6. 正整數的分割 (partition) 與組成 (arrangements) (1 週)
  7. 排容原理 (1 週)
  8. 排列的統計量 (2 週)
  9. 學生專題報告 (8 週)

註：學生將需要使用數學軟體 SAGE (python based) 或其他程式語言編寫程式。本課程中一定比例以英語授課。

 

The sessions are arranged as follows:

1. Basic counting principles: rules of addition and multiplication, bijections (1 week)

2. Binomial coefficients (2 weeks)

3. Generating functions and recursions (1 week)

4. Set partition and the Stirling numbers of the second kind (1 week)

5. Permutations and the Stirling numbers of the first kind (1 week)

6. Partitions and arrangements (1 week)

7. Principle of inclusion-exclusion (1 week)

8. Statistics on permutations (2 weeks)

9. Student reports

Miscellaneous:

1. Students are required to code in SAGE (a python-based mathematical software)

2. English will be used in this class
Instructional Approach
Methods Notes
Formal lecture 上課講解
Problem-based learning 提出問題，介紹解決工具
Media, audio, visual materials 編寫程式，由結果推測公式
Case studies 學生專題報告
Grading assessment
Methods Percentage Notes
Midterm Exam 40 % 預備基本知識
Attendances 10 % 準時出席討論
Case study reports 50 % 期末專題報告
Required and Recommended Texts/Readings with References
  1. Alan Tucker, Applied Combinatorics, 6th edition, Wiley
  2. Richard Brualdi, Introductory Combinatoric, 4th edition, Pearson
  3. Martin Aigner, A Course in Enumeration, Graduate Texts in Mathematics 238, Springer-Verlag
  4. Richard Stanley, Enumerative Combinatorics, vols. 1 & 2, Cambridge
  5. Ronald Graham et al., Concrete Mathematics, 2nd edition, Addison-Wesley

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