Stirling Numbers of the Second Kind S(n,k)

Compute the Stirling number of the second kind S(n,k), the number of ways to partition n elements into k non-empty subsets, with the related Bell number.

Input

Enter n (number of elements) and k (number of subsets) to compute the Stirling number of the second kind S(n,k): the number of ways to split n distinct elements into k non-empty subsets.

n (elements)

Integer from 0 to 200

k (subsets)

Non-negative integer

Result

S(5, 2) — partitions into 2 non-empty subsets

Bell number B(5) (row sum)

Elements n

Subsets k

Row for n = 5: S(n,0) to S(n,n)

S(n, k)	Value
S(5, 0)	0
S(5, 1)	1
S(5, 2)	15
S(5, 3)	25
S(5, 4)	10
S(5, 5)	1

How it works

The Stirling number of the second kind S(n,k) counts the ways to partition a set of n distinct elements into k non-empty, unlabeled subsets.
It is computed with the recurrence S(n,k) = k times S(n-1,k) plus S(n-1,k-1): the new element either joins one of the existing k subsets or forms a brand-new singleton subset.
Boundary conditions are S(0,0) = 1, S(n,0) = 0 for n greater than 0, and S(0,k) = 0 for k greater than 0. When k exceeds n the value is 0.
The row sum S(n,0) + S(n,1) + ... + S(n,n) equals the Bell number B(n), the total number of partitions of an n-element set.
Because the values grow very fast, the calculation uses arbitrary-precision integers (BigInt) for exact results, displayed as strings.

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