What do Stirling numbers of the second kind count?

S(n, k) counts the number of ways to partition a set of n distinct objects into exactly k non-empty, unlabeled subsets.

What do Stirling numbers of the first kind count?

The unsigned Stirling number c(n, k) counts the number of permutations of n elements that have exactly k cycles when written in cycle notation.

What is the difference between signed and unsigned first-kind Stirling numbers?

The unsigned value c(n, k) is always non-negative and has the direct permutation-cycle-counting meaning; the signed value s(n, k) = (−1)ⁿ⁻ᵏ × c(n, k) appears when Stirling numbers are used as coefficients relating factorial powers to ordinary powers.

What is the recurrence for the second kind?

S(n, k) = k × S(n−1, k) + S(n−1, k−1), with base cases S(0, 0) = 1 and S(n, 0) = 0 for n > 0.

What is the recurrence for the first kind?

c(n, k) = (n−1) × c(n−1, k) + c(n−1, k−1), with the same style of base cases — reflecting that adding the nth element either becomes its own cycle or is inserted into one of the existing n−1 positions.

What happens if k > n?

Both kinds of Stirling numbers are 0 whenever k > n, since you can't partition n objects into more than n non-empty groups (or form more than n cycles from n elements).

This calculator supports n up to 500, using a dynamic-programming table computed with exact BigInt arithmetic so results stay exact no matter how many digits they have.

Where are Stirling numbers used in real life?

They appear in combinatorics (counting partitions and permutation structures), the analysis of algorithms (especially those involving permutations), and in expressing polynomials in terms of falling or rising factorials.

Stirling Numbers Calculator — Compute S(n,k) and c(n,k)

Our Stirling Numbers Calculator computes both kinds of Stirling numbers: the second kind S(n, k), counting ways to partition n objects into k non-empty subsets, and the first kind c(n, k) (unsigned) or s(n, k) (signed), counting permutations of n elements arranged into k cycles — both built up via their standard recurrences with exact BigInt precision.

Quick Answer

Choose the first or second kind, enter n and k, and this calculator computes the exact Stirling number via its standard recurrence, with BigInt precision.

Enter n and k, then click Calculate.

How to Use the Stirling Numbers Calculator — First & Second Kind Online

1
Choose the first kind or second kind of Stirling number.
2
Enter n and k (both non-negative whole numbers, k ≤ n).
3
Click 'Calculate' to see the exact value, computed via the standard recurrence.

Why Use Stirling Numbers Calculator — First & Second Kind Online?

Stirling numbers, named after James Stirling, come in two families that both decompose a combinatorial count by recursively considering the last element: the second kind S(n, k) counts ways to partition n labeled objects into k non-empty unlabeled groups, while the first kind counts permutations of n elements by their number of cycles. Both satisfy a clean recurrence relating S(n, k) (or c(n, k)) to smaller cases, which this calculator applies directly with a dynamic-programming table and exact BigInt arithmetic, since both families grow extremely quickly.

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