18.314 | Fall 2014 | Undergraduate

# Combinatorial Analysis

Assignments

## Problem Set 3

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 (+)

• (A2) Let λ be a partition with conjugate λ’. Show that

Σ i ⌊λ2_i-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.

• (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.
• Bonus Problems

• None

## Course Info

Fall 2014
##### Learning Resource Types
Exams with Solutions
Problem Sets