18.314 | Fall 2014 | Undergraduate
Combinatorial Analysis
Assignments

## Problem Set 8

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 8

• Due in Session 22
• Practice Problems
• Session 19: None from textbook
• A group of n children form circles by holding hands, with a child in the center of each circle. Let h(n) be the number of ways that this can be done. Set h(0)=1. Find a simple expression for the generating function F(x) = Σn≥₀h(n)xⁿ/n!. A circle may consist of just one child holding his or her own hands, but a child must be in the center of each circle. The clockwise order of the children around the circle matters, so k children can form a circle in (k-1)! ways. Thus h(1)=0, h(2)=2, h(3)=3, h(4)=20, h(5)=90. Answer: (1-x
• Session 20: Chapter 9: Exercises 1, 2, 3, 6, 16, 18
• Session 21: Chapter 9: Exercises 8, 11, 14, 21
• Problems Assigned in the Textbook
• Chapter 9: Exercises 24, 30
• Chapter 9: Exercises 34, 41. Hint for 41: Induction on n.