Stirling Number Calculator

Calculate Stirling numbers of the first and second kind, plus Bell numbers for set partitions.

Kind (1=First, 2=Second)

Tab to move between fields

Stirling Number S(n,k)

Bell Number B(n)

How to Use This Calculator

Enter n and k — the Stirling number S(n, k) counts a specific combinatorial structure.
Select Kind: First (signed cycle decompositions) or Second (partitions into non-empty subsets).
Review the Stirling Number S(n, k) for the selected kind.
Check Bell Number B(n) — the sum of all Stirling numbers of the second kind for a given n.
Use Stirling numbers in combinatorics, probability, and the analysis of divide-and-conquer algorithms.

How the result changes with n

n: Size of the set (max 25 to avoid overflow)
k: Number of cycles (1st kind) or blocks/partitions (2nd kind)
Kind (1=First, 2=Second): Stirling number kind: 1st (permutation cycles) or 2nd (set partitions)

Identify Input Parameters
n = 5, k = 3, Kind (1=First, 2=Second) = 2 = 3 input(s) provided
Calculate Stirling Number S
Stirling Number S
25 = 25
Calculate Bell Number B
Bell Number B
52 = 52

Engine last updated August 5, 2026. Checked against 7 independently-derived tests — how we verify calculators.

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