18.314 | Fall 2014 | Undergraduate
Combinatorial Analysis
Assignments

Problem Set 7

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.
    • Session 18: Chapter 8: Exercises 20, 21
  • 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-qx²)(1-qx³)…
  • 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.
  • Bonus Problems
    • (B2) Find the generating function G(x) = Σ_n_≥0 anxn/n!,    
      where an+1 = (n+1)an-{n\choose 2}an-2 for n≥0, and a0=1.     
      Thus a1=1, a2=2, a3=5. You don’t need to find a formula for an.
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Fall 2014
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assignment Problem Sets