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Stirling Numbers Calculator

Stirling numbers of the first (signed & unsigned) and second kind, S(n,k), by exact recurrence.

Related tools

All Combinatorics tools →
Cube numbers listGenerate a list of perfect cubes — n³ for consecutive integers from any starting point. Set how many you want and where to start, and it returns the cubes, their sum and the underlying roots, computed with exact big-integer arithmetic.First n prime numbersList the first n prime numbers — the building blocks of arithmetic that have no divisors but 1 and themselves. Enter how many you want (up to 10,000) and it returns the sequence, the nth prime itself and the sum of them all.List of Fibonacci numbersGenerate the first n Fibonacci numbers exactly, with big-integer precision — each term the sum of the two before it, starting 0, 1, 1, 2, 3, 5, 8… The sequence hides the golden ratio and shows up throughout nature and mathematics. Returns the list and its sum.List of prime numbersList every prime number up to a chosen limit, using the Sieve of Eratosthenes. Enter an upper bound (up to 100,000) and it returns all primes at or below it, how many there are and their sum — a quick reference for factoring and number theory.Square numbers listGenerate a list of perfect squares — n² for consecutive integers from any starting point. Choose how many and where to begin, and it returns the squares and their sum with exact big-integer arithmetic, handy for spotting patterns and differences.Combination calculator (nCr)Compute the number of combinations nCr of r items from n.Derangement (subfactorial) calculator !nCompute the subfactorial !n — the number of derangements, permutations of n items in which nothing stays in its original place. It uses the exact recurrence !n = (n−1)·(!(n−1) + !(n−2)) with big integers, and shows the ratio !n / n!, which converges astonishingly fast to 1/e ≈ 0.3679. That ratio is the probability that a random shuffle leaves no element fixed — the classic 'hat-check' problem.Factorial calculatorCompute the factorial n! of a whole number.

Stirling Numbers Calculator is free to use as often as you like, directly from this page. You will find it under Combinatorics, with Cube numbers list and First n prime numbers for the neighbouring cases.

How to use it

  1. Open the tool — no signup or install needed.
  2. Enter your input or adjust the available options.
  3. Get your result instantly, then copy or download it.

Frequently asked questions

What is Stirling Numbers Calculator?

Stirling numbers of the first (signed & unsigned) and second kind, S(n,k), by exact recurrence.

When would I actually use this?

Counting possibilities before enumerating them: hands in a card game, passwords of a given shape, seatings around a table, or lottery odds.

What is the most common mistake?

Choosing a permutation when order does not matter. Picking three people from ten gives 720 arrangements but only 120 groups — the two differ by a factor of six here, and far more as the numbers grow.

How is Stirling Numbers Calculator different from Cube numbers list?

They sit next to each other but answer different questions: Cube numbers list is the one to open when you need it to generate a list of perfect cubes — n³ for consecutive integers from any starting point. Set how many you want and where to start, and it returns the cubes, their sum and the underlying roots, computed with exact big-integer arithmetic. Pick whichever matches what you're starting from — both are free.

Is there a tool for the next step?

First n prime numbers is the closest one after this: List the first n prime numbers — the building blocks of arithmetic that have no divisors but 1 and themselves. Enter how many you want (up to 10,000) and it returns the sequence, the nth prime itself and the sum of them all.

What else is worth having open alongside it?

List of Fibonacci numbers and List of prime numbers — they come up in the same task often enough to be worth a second tab.

Where do the figures come from?

Counting formulas are exact by definition. Large factorials are computed with arbitrary precision where needed, so a result is not silently rounded into scientific notation.

Further reading

All guides →
ExplainerWhat Is a Factorial? n! Explained SimplyA factorial multiplies every whole number down to 1. Learn what n! means, how fast it grows, and why it powers permutations and combinations.ExplainerCombinations vs Permutations: When Order MattersThe difference between combinations and permutations comes down to one question: does order matter? Learn the nCr and nPr formulas with clear worked examples.ExplainerOrdinal Numbers Explained: 1st, 2nd, 3rdOrdinal numbers show position, cardinal numbers show quantity. Learn the difference and how each language forms ordinals differently.ExplainerHow Exponents Work: Powers and the Laws of ExponentsAn exponent is repeated multiplication. Learn the laws of exponents, what negative and fractional powers mean, and why anything to the zero is one.GuideTournament Brackets: Byes, Seeds, and Why the Numbers Must Be Powers of TwoA knockout bracket halves its field every round, so it only closes on a power of two. The bye count, the round count, the seed order and the total number of matches all follow from that one fact — and each is a one-line derivation.ExplainerPoker Hand Probabilities, Derived Rather Than MemorisedEvery five-card poker probability is a counting argument over 2,598,960 hands, and each one fits on a single line. Here are all nine, with the check that proves them: the counts must sum to exactly C(52,5).
Stirling Numbers Calculator — OneKitly