Exercise · ZW-INTEGER-PARTITION-008 · ★★☆ Intermediate
What is the coefficient of \(x^3\) in \(\prod_{k\ge 1}\frac{1}{1-x^k}\)?
Choose one option.
Show answer
- D. \(3\)
Explanation
The coefficient of \(x^3\) equals p(3), which is 3 from the partitions 3, 2+1 and 1+1+1. The option giving five corresponds to p(4), not p(3).
Review the concept: Integer partition
A partition of an integer n is a way of writing n as a sum of positive integers, where the order of the summands does not matter. The number of partitions of n is denoted p(n).
\[\sum_{n\ge 0} p(n)x^n=\prod_{k\ge 1}\frac{1}{1-x^k}\]
More practice problems: integer partition
- ZW-INTEGER-PARTITION-001 How many partitions does the integer 3 have?
- ZW-INTEGER-PARTITION-002 Which of the following is a partition of 5?
- ZW-INTEGER-PARTITION-003 How many partitions of 4 have exactly two parts?
- ZW-INTEGER-PARTITION-004 What is the number of partitions p(2)?
- ZW-INTEGER-PARTITION-005 Which partition of 6, written in non-increasing order, has all its parts equal to 2?
- ZW-INTEGER-PARTITION-006 How many partitions does 5 have?
- ZW-INTEGER-PARTITION-007 How many partitions of 6 have all parts distinct?
- ZW-INTEGER-PARTITION-009 How many partitions of 4 have all parts odd?