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 7

- Due in Session 19
- Practice Problems
- Session 17: None from textbook
- Let
*n*≥1, and let*f*(*n*) be the number of partitions of*n*such that for all*k*, the part*k*occurs at most*k*times. Let*g*(*n*) be the number of partitions of*n*such that no part has the form*i*(*i*+1), i.e., no parts equal to 2, 6, 12, 20, …. Show that*f*(*n*)=*g*(*n*). Use generating functions. - Let
*f*(*n*) denote the number of partitions of*n*with an even number of 1's. Give a combinatorial proof and a generating function proof that*f*(*n*) +*f*(*n*-1) =*p*(*n*), the total number of partitions of*n*.

- Let
- Session 18: Chapter 8: Exercises 20, 21

- Session 17: None from textbook
- Problems Assigned in the Textbook
- Chapter 8: Exercises 27, 28, 32, 37*. In exercise 28, you can ignore the last sentence (about comparing with Exercise 4).
**Hint**for 37. Consider the product 1/(1-*qx*)(1-*qx*^{2})(1-*qx*^{3})...

- Chapter 8: Exercises 27, 28, 32, 37*. In exercise 28, you can ignore the last sentence (about comparing with Exercise 4).
- Additional Problems
- (A8*) Show that the number of partitions of
*n*for which no part appears exactly once is equal to the number of partitions of*n*for which every part is divisible by 2 or 3. For instance, when*n*=6 there are four partitions of the first type (111111,2211,222,33) and four of the second type (222, 33, 42, 6). Use generating functions. - (A9) Show that the number of partitions of
*n*for which no part appears more than twice is equal to the number of partitions of*n*for which no part is divisible by 3. For instance, when*n*=5 there are five partitions of the first type (5, 41, 32, 311, 221) and five of the second type (5, 41, 221, 2111, 11111). Use generating functions.

- (A8*) Show that the number of partitions of
- Bonus Problems
- (B2) Find the generating function
*G*(*x*) = Σ_{n≥0}an*x*_{n}/*n*!,

where*a*_{n+1}= (*n*+1)*a*_{n}-{*n*\choose 2}*a*_{n-2}for*n*≥0, and*a*_{0}=1.

Thus*a*_{1}=1,*a*_{2}=2,*a*_{3}=5. You don't need to find a formula for*a*_{n}.

- (B2) Find the generating function