Most of the problems are assigned from the required textbook Bona, Miklos. A Walk Through Combinatorics: An Introduction to Enumeration and Graph Theory. World Scientific Publishing Company, 2011. ISBN: 9789814335232. [Preview with Google Books]
A problem marked by * is difficult; it is not necessary to solve such a problem to do well in the course.
Problem Set 3
- Due in Session 8
- Practice Problems
- Session 6: Chapter 5: Exercises 11, 12, 13
- Session 7: Chapter 5: Exercises 1, 5, 16
- Problems Assigned in the Textbook
- Chapter 5: Exercise 21
- Chapter 5: Exercise 34. Only do the case k=1, which is already pretty tricky and in my opinion deserves a (+)
- Additional Problems
- (A2) Let λ be a partition with conjugate λ'. Show that
- (A3) Show by simple combinatorial reasoning and induction that the Bell number B(n) is even if and only if n-2 is divisible by 3.
Σ i ⌊λ2i-1/2⌋ = Σ i ⌈λ'2i/2⌉.
This can be seen almost by inspection from the Young diagram of λ after certain marks are made on it. Note. The notation ⌊x⌋ means the greatest integer ≤x. For instance, ⌊3⌋=3, ⌊3/2⌋=1. Similarly ⌈x⌉ means the least integer ≥x. For instance, ⌈3⌉=3, ⌈3/2⌉=2.
- Bonus Problems