18.314 | Fall 2014 | Undergraduate
Combinatorial Analysis
Assignments

Problem Set 3

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 3

  • Due in Session 8

  • Practice Problems

    • Session 6: Chapter 5: Exercises 11, 12, 13
    • Session 7: Chapter 5: Exercises 1, 5, 16
  • Problems Assigned in the Textbook

    • Chapter 5: Exercise 21
    • Chapter 5: Exercise 34. Only do the case k=1, which is already pretty tricky and in my opinion deserves a (+)
  • Additional Problems

    • (A2) Let λ be a partition with conjugate λ’. Show that

    Σ i ⌊λ2_i-1_/2⌋ = Σ i ⌈λ’2i/2⌉.
    This can be seen almost by inspection from the Young diagram of λ after certain marks are made on it. Note. The notation ⌊x⌋ means the greatest integer ≤_x_. For instance, ⌊3⌋=3, ⌊3/2⌋=1. Similarly ⌈x⌉ means the least integer ≥_x_. For instance, ⌈3⌉=3, ⌈3/2⌉=2.

    • (A3) Show by simple combinatorial reasoning and induction that the Bell number B(n) is even if and only if n-2 is divisible by 3.
  • Bonus Problems

    • None
Course Info
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As Taught In
Fall 2014
Learning Resource Types
grading Exams with Solutions
assignment Problem Sets