Partition function p(n): ways to write n as sum of positive integers (order doesn't matter). p(4)=5: 4, 3+1, 2+2, 2+1+1, 1+1+1+1. Generating function: Π1/(1−xᵏ). Ramanujan: p(5n+4)≡0(mod 5), p(7n+5)≡0(mod 7), p(11n+6)≡0(mod 11).
How to Use
Enter n:
- n: Positive integer
- p(n): Number of partitions
- List: All partitions
Properties
- p(n) ~ exp(π√(2n/3))/(4n√3) (Hardy-Ramanujan)
- Euler's pentagonal theorem gives recurrence
- Conjugate partitions: transpose Ferrers diagram
Ramanujan Congruences
p(5n+4)≡0(mod 5). p(7n+5)≡0(mod 7). p(11n+6)≡0(mod 11). Example: p(4)=5, p(9)=30, p(14)=135 — all divisible by 5.
Step-by-Step Instructions
- 1Enter n.
- 2Get p(n).
- 3List partitions.
- 4Check congruences.
- 5View table.