Problem Set 8

Most of the problems are assigned from the required textbook Amazon logo 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≥0 h(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)-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.
  • 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) = Σn≥0 f(n)xn/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.
  • Bonus Problems
    • Chapter 9: Exercise 49