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}⌊λ2/2⌋ = Σ_{i-1}_{i }⌈λ'/2⌉._{2i}

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
- None