Partition numbers p(n): ways to write n as unordered sums. p(4)=5: 4, 3+1, 2+2, 2+1+1, 1+1+1+1. Generating function: Σp(n)x^n = Π_{k≥1}1/(1-x^k). Hardy-Ramanujan: p(n)~exp(π√(2n/3))/(4n√3). Ramanujan congruences: p(5n+4)≡0 mod 5.
How to Use
Enter n:
- p(n): Total partitions
- List: All partitions
- Distinct: Parts all different
Ramanujan
Remarkable congruences: p(5n+4)≡0(mod 5), p(7n+5)≡0(mod 7), p(11n+6)≡0(mod 11). The Hardy-Ramanujan-Rademacher formula gives an EXACT series for p(n), one of the most beautiful results in analytic number theory.
Applications
- Statistical mechanics (Bose-Einstein)
- Representation theory (symmetric group)
- Algebraic geometry (Hilbert schemes)
- Combinatorics (q-series)
Step-by-Step Instructions
- 1Enter n.
- 2Compute p(n).
- 3List partitions.
- 4Check Hardy-Ramanujan.
- 5See distinct.