Integer partitions of n: all distinct ways to write n as sum of positive integers (order doesn't matter). p(5)=7: 5=4+1=3+2=3+1+1=2+2+1=2+1+1+1=1+1+1+1+1. p(n) grows exponentially: p(100)=190,569,292,356. Hardy-Ramanujan formula: p(n) ~ exp(π√(2n/3))/(4n√3).
How to Use
Enter n:
- Count: p(n) total
- List: All partitions
- Diagrams: Ferrers/Young
Ferrers Diagrams
Each partition maps to a dot diagram: row i has kᵢ dots. The conjugate partition is the transpose. Ferrers diagrams reveal symmetries and prove identities (e.g., # partitions into ≤k parts = # partitions with largest part ≤k).
Ramanujan
Ramanujan discovered remarkable congruences: p(5n+4)≡0 (mod 5), p(7n+5)≡0 (mod 7), p(11n+6)≡0 (mod 11). The Hardy-Ramanujan-Rademacher exact formula gives p(n) as a convergent series.
Step-by-Step Instructions
- 1Enter n.
- 2See count.
- 3View all partitions.
- 4Explore diagrams.
- 5Check conjugates.