Exercise · ZW-PARTITION-GENERATING-FUNCTION-008 · ★★☆ Intermediate
What is the coefficient of \(x^5\) in \(\frac{1}{(1-x)(1-x^2)}\)?
Choose one option.
Show answer
- C. 3
Explanation
The ways to write 5 with parts 1 and 2 are \(2+2+1\), \(2+1+1+1\), and \(1+1+1+1+1\), three in total. Leaving one out gives 2, and adding a nonexistent way gives 4.
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-009 How many partitions of 4 have all their parts equal to 1 or 2?
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