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*) = Σ_{n≥0}*h*(*n*)*x*/^{n}*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*)^{-}^{x}

- A group of
- Session 20: Chapter 9: Exercises 1, 2, 3, 6, 16, 18
- Session 21: Chapter 9: Exercises 8, 11, 14, 21

- Session 19: None from textbook
- Problems Assigned in the Textbook
- Chapter 9: Exercises 24, 30
- Chapter 9: Exercises 34, 41.
**Hint**for 41: Induction on*n*.

- Additional Problems
- (A10) Let
*f*(*n*) be the number of ways to paint*n*giraffes either red, blue, yellow, or turquoise, such that an odd number of giraffes are red and an even number are blue. Use exponential generating functions to find a simple formula for*f*(*n*). (It is allowed to have no giraffes painted blue, yellow, or turquoise.) - (A11) Let
*f*(*n*) be the number of ways to partition an*n*-element set, and then to choose a nonempty subset of each block of the partition. Find a simple formula (no infinite sums) for the exponential generating function*G*(*x*) = Σ_{n≥0}*f*(*n*)*x*/^{n}*n*!. - (A12) Give a simple reason why a 9-vertex simple graph cannot have the degrees of its vertices equal to 8, 8, 6, 5, 5, 4, 4, 3, 1.

- (A10) Let
- Bonus Problems
- Chapter 9: Exercise 49