What is a Stirling number of the first kind?

The unsigned Stirling number of the first kind c(n, k) counts the number of permutations of n elements with exactly k disjoint cycles. The signed version s(n, k) = (-1)^(n-k) * c(n, k) appears when expressing falling factorials as polynomials.

What is a Stirling number of the second kind?

S(n, k) counts the number of ways to partition a set of n elements into exactly k non-empty, non-overlapping subsets. For example, S(4, 2) = 7 means there are 7 ways to split 4 elements into 2 non-empty groups.

What is the recurrence relation for Stirling numbers?

First kind (unsigned): c(n, k) = (n-1)*c(n-1, k) + c(n-1, k-1). Second kind: S(n, k) = k*S(n-1, k) + S(n-1, k-1). Both use boundary conditions c(0,0)=S(0,0)=1 and c(n,0)=S(n,0)=0 for n>0.

How large can n and k be?

This calculator supports n and k up to 20. Stirling numbers grow very rapidly — S(20, 10) exceeds 10^13 — so JavaScript 64-bit floats handle values up to n≈18 exactly. For n=19 or 20, results are rounded to the nearest integer and may lose the last digit of precision.

What is the relationship between Stirling numbers and Bell numbers?

The Bell number B(n) counts all partitions of an n-element set. It equals the sum of S(n, k) for k from 0 to n. This calculator shows each row sum in the second-kind table.

Stirling Number Calculator

Name: Stirling Number Calculator
Availability: InStock
Author: Nham Vu

Calculate Stirling numbers of the first and second kind for any n and k values up to 20.

Parameters

Type

Unsigned: c(n, k) — permutations of n elements with k cycles.

Variant

n (total elements, 0–20)

k (groups / cycles, 0–20)

Result

c(5, 2)

Triangle Table

rows 0–5

Each cell shows the value for that (n, k) pair. The highlighted cell is your selected (n, k).

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Summary

Calculate Stirling numbers of the first and second kind for any n and k values up to 20.

How it works

Select whether you want Stirling numbers of the first kind (cycle permutations) or the second kind (set partitions).
Enter the values of n (number of elements) and k (number of groups/cycles).
Click Calculate to compute the result using the triangular recurrence relation.
View the full triangle table up to n for quick lookup of related values.
Copy the result to the clipboard for use in your work.

Use cases

Count the number of ways to arrange n objects into k non-empty cycles (first kind).
Count the number of ways to partition n objects into k non-empty subsets (second kind).
Verify combinatorics homework or exam answers for Stirling number problems.
Explore the Stirling triangle table for pattern recognition in combinatorics research.
Convert between falling factorials and ordinary polynomials using first kind coefficients.
Compute Bell numbers by summing Stirling numbers of the second kind across k.

Frequently Asked Questions

Last updated: 2026-06-13 · Reviewed by Nham Vu