18.314 | Fall 2014 | Undergraduate
Combinatorial Analysis
Assignments

## Problem Set 6

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≤in 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)xn. 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.
• 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 Cn. Give a bijection with ballot sequences of length 2_n_. (Ballot sequences and Catalan numbers were discussed in class.)
• 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, Hn should be hn. Also find a simple explicit formula for hn.