Stirling numbers: S₂(n,k) = ways to partition {1,...,n} into k non-empty subsets. Recurrence: S₂(n,k) = k·S₂(n−1,k) + S₂(n−1,k−1). Bell number: Bₙ = Σₖ S₂(n,k). First kind |s₁(n,k)|: permutations of n with exactly k cycles.
How to Use
Enter parameters:
- n: Total elements
- k: Number of parts/cycles
- Output: S(n,k)
Second Kind
S₂(n,k) = (1/k!)Σⱼ(−1)ʲC(k,j)(k−j)ⁿ. Base: S₂(n,1)=1, S₂(n,n)=1. S₂(n,2)=2ⁿ⁻¹−1. xⁿ = Σₖ S₂(n,k)·x(x−1)···(x−k+1).
First Kind
|s₁(n,k)| counts permutations with k cycles. s₁(n,1) = (n−1)!. x(x+1)···(x+n−1) = Σₖ |s₁(n,k)|xᵏ (rising factorial).
Step-by-Step Instructions
- 1Enter n and k.
- 2Get S₂(n,k).
- 3Get |s₁(n,k)|.
- 4View triangle.
- 5Compute Bell number.