Exercise · ZW-PARTITION-GENERATING-FUNCTION-010 · ★★★ Advanced
Let \(F(x)=\prod_{k\ge1}\frac{1}{1-x^k}\). What is the coefficient of \(x^5\) in \(F(x)\)? Hint: \(p(5)\) comes from counting all partitions of 5.
Choose one option.
Show answer
- A. 7
Explanation
The partitions of 5 are seven: \(5\), \(4+1\), \(3+2\), \(3+1+1\), \(2+2+1\), \(2+1+1+1\), and \(1+1+1+1+1\). Omitting one gives 6, and adding an improper composition gives 8.
Review the concept: Partition generating function
The partition generating function is the formal series \(\sum_{n\ge 0} p(n)\,x^n\), where \(p(n)\) counts the partitions of the integer \(n\).
\[\sum_{n\ge 0} p(n)x^n=\prod_{k\ge 1}\frac{1}{1-x^k}\]
More practice problems: partition generating function
- ZW-PARTITION-GENERATING-FUNCTION-001 What is the coefficient of x^1 in prod_k≥1(1)/(1-x^k)?
- ZW-PARTITION-GENERATING-FUNCTION-002 How many partitions does the integer 2 have?
- ZW-PARTITION-GENERATING-FUNCTION-003 Which coefficient in the expansion of prod_k≥1(1)/(1-x^k) equals p(3)?
- ZW-PARTITION-GENERATING-FUNCTION-004 What factor does the part of size k contribute to the generating function?
- ZW-PARTITION-GENERATING-FUNCTION-005 What is the coefficient of x^2 in prod_k≥1(1)/(1-x^k)?
- ZW-PARTITION-GENERATING-FUNCTION-006 If p(n) is the number of partitions of n, what is p(3)?
- ZW-PARTITION-GENERATING-FUNCTION-007 Using 1+x+2x^2+3x^3+5x^4+cdots, what is p(4)?
- ZW-PARTITION-GENERATING-FUNCTION-008 What is the coefficient of x^5 in (1)/((1-x)(1-x^2))?