Partition Number Calculator
Calculate p(n), the number of ways to write n as a sum of positive integers.
Calculate p(n)
p(10)
42
All partitions
q(10)
10
Distinct parts only
Odd parts
10
= q(n) (Euler)
Partition Sequence
p(0)=1p(1)=1p(2)=2p(3)=3p(4)=5p(5)=7p(6)=11p(7)=15p(8)=22p(9)=30p(10)=42
Partitions of 10
10
9 + 1
8 + 2
8 + 1 + 1
7 + 3
7 + 2 + 1
7 + 1 + 1 + 1
6 + 4
6 + 3 + 1
6 + 2 + 2
6 + 2 + 1 + 1
6 + 1 + 1 + 1 + 1
5 + 5
5 + 4 + 1
5 + 3 + 2
5 + 3 + 1 + 1
5 + 2 + 2 + 1
5 + 2 + 1 + 1 + 1
5 + 1 + 1 + 1 + 1 + 1
4 + 4 + 2
4 + 4 + 1 + 1
4 + 3 + 3
4 + 3 + 2 + 1
4 + 3 + 1 + 1 + 1
4 + 2 + 2 + 2
4 + 2 + 2 + 1 + 1
4 + 2 + 1 + 1 + 1 + 1
4 + 1 + 1 + 1 + 1 + 1 + 1
3 + 3 + 3 + 1
3 + 3 + 2 + 2
3 + 3 + 2 + 1 + 1
3 + 3 + 1 + 1 + 1 + 1
3 + 2 + 2 + 2 + 1
3 + 2 + 2 + 1 + 1 + 1
3 + 2 + 1 + 1 + 1 + 1 + 1
3 + 1 + 1 + 1 + 1 + 1 + 1 + 1
2 + 2 + 2 + 2 + 2
2 + 2 + 2 + 2 + 1 + 1
2 + 2 + 2 + 1 + 1 + 1 + 1
2 + 2 + 1 + 1 + 1 + 1 + 1 + 1
2 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1
1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1
Properties
Euler's Identity
Number of partitions into distinct parts = partitions into odd parts
Generating Function
Σp(n)x^n = Π 1/(1-x^k)
Hardy-Ramanujan
p(n) ~ exp(π√(2n/3)) / (4n√3)
Pentagonal Theorem
p(n) = Σ (-1)^(k+1) p(n - k(3k-1)/2)