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 6

- Due in Session 17
- Practice Problems
- Session 14: Chapter 8: Exercises 1, 2, 4, 7, 8, 10, 14*
- Additional practice problem: Let
*f*(*n*) denote the number of permutations*π*of 1,2,…,*n*such that for all 1≤*i*≤*n*we have π(*i*) =*i*-1,*i*,*i*+1, or*i*+2. (Set*f*(0)=1.) For instance,*f*(3)=4, the four permutations being 123, 132, 213, 312. Find the generating function*G*(*x*) = Σ_{_n_≥0}f(*n*)*x*_{n}. You do not need to find a formula for*f*(*n*). What if we do not allow fixed points, i.e., we exclude π(*i*)=*i*?**Hint.**Consider the digraph of π where we write the vertices 1,2,…,n in a line.

- Additional practice problem: Let
- Session 15: None from textbook
- (additional practice problem): Suppose we have 2_n_ points on the circumference of a circle. Show that the number of ways we can draw
*n noncrossing*diagonals connecting the points, such that each of the points is an endpoint of one of the diagonals, is equal to the Catalan number*C*_{n}. Give a bijection with ballot sequences of length 2_n_. (Ballot sequences and Catalan numbers were discussed in class.)

- (additional practice problem): Suppose we have 2_n_ points on the circumference of a circle. Show that the number of ways we can draw

- Session 14: Chapter 8: Exercises 1, 2, 4, 7, 8, 10, 14*
- Problems Assigned in the Textbook
- Chapter 8: Exercises 24, 26, 35. In 24 use generating functions. Do not simply guess the answer and verify that it is correct. In 35,
*H*_{n}should be*h*_{n}. Also find a simple explicit formula for*h*_{n}.

- Chapter 8: Exercises 24, 26, 35. In 24 use generating functions. Do not simply guess the answer and verify that it is correct. In 35,
- Additional Problems
- (A7) Let
*f*(*n*) be the number of ways to stack pennies against a flat wall as follows: the bottom level consists of a row of*n*pennies, each tangent to its neighbor(s). A penny may be placed in a higher row if it is supported by two pennies below it. Here is an example (PDF) for*n*=10 and for all five possibilities when*n*=3. Show that*f*(*n*)=*C*_{n}, a Catalan number. (One of many methods is to give a bijection with the Dyck paths discussed in class.)

- (A7) Let
- Bonus Problems
- None