Bell number Bₙ: total partitions of {1,...,n}. B₀=1, B₁=1, B₂=2, B₃=5, B₄=15, B₅=52. Bₙ = Σₖ S₂(n,k) (sum of Stirling second kind). Dobinski: Bₙ = (1/e)Σₖ₌₀ kⁿ/k!. Bell triangle recurrence for computation.
How to Use
Enter n:
- n: Set size
- Bₙ: Bell number
- Triangle: Bell triangle
Bell Triangle
Row 0: [1]. Each row starts with last element of previous row, then aᵢ = aᵢ₋₁ + element above aᵢ₋₁. Bₙ = first element of row n+1 = last element of row n.
Applications
- Counting equivalence relations
- Set partition lattice
- Exponential generating function: e^(eˣ−1)
- Combinatorial species
Step-by-Step Instructions
- 1Enter n.
- 2Get Bₙ.
- 3View Bell triangle.
- 4See Stirling decomposition.
- 5Check table.