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COMBINATORICS / EXACT BLOCK COUNT

Stirling Number Second Kind Calculator

Count partitions of n labelled elements into exactly k nonempty unlabeled blocks.

  • 01 Calculated in this tab
  • 02 Values stay in this browser tab
  • 03 Use boundary

Conversion input

Known value

Filter by unit name, symbol, or code. Your current selections remain available.

Exact integer inputs

Preparing the calculator...

METHOD / WORKED EXAMPLE

Verify Stirling Number Second Kind from its integer structure

Count partitions of n labelled elements into exactly k nonempty unlabeled blocks. The page exposes the recurrence, factorization, or residue evidence used to obtain the bounded exact result.

WORKED DEFAULT

Check the calculation with the default inputs

For n = 10 labelled elements and k = 3 nonempty blocks, the recurrence gives S(10,3) = 9,330.

  1. Set the staten = 10; k = 3
  2. Apply recurrencek S(n-1,k) + S(n-1,k-1)
  3. Read exact countS(10,3) = 9,330

READ THE RESULT

Interpret the output in context

Blocks are nonempty and unlabeled while elements are distinct; changing any of those conventions changes the count.

ASSUMPTIONS AND LIMITS

Know where the model stops

  • Zero through twenty are accepted with k no greater than n.
  • S(0,0)=1 and impossible boundary states equal zero.

The bounds preserve exact safe integers and predictable dynamic-programming work in the browser.

NIST DLMF: Stirling numbers

COMMON QUESTIONS

Stirling Number Second Kind Calculator FAQs

Which definition does Stirling Number Second Kind Calculator use?

Use the second-kind recurrence: place the newest labelled element into one of k existing blocks or make it the first member of a new block. The second kind counts set partitions; signed and unsigned first-kind Stirling numbers count permutation cycles instead. Inputs must be whole numbers inside the displayed safe bound. The calculator does not round decimals into the domain or silently substitute a different convention, so verify each entered integer before interpreting the exact result.

How can I check the Stirling Number Second Kind result?

Blocks are nonempty and unlabeled while elements are distinct; changing any of those conventions changes the count. Compute neighboring previous-row values and substitute them into the displayed recurrence to check a term. Use the displayed factors, recurrence step, or modular residue as an independent check. Exact integer output means the implemented arithmetic has no decimal approximation within its bound; it does not prove an unrelated theorem or an unstated combinatorial model.

Why does this calculator impose a finite bound?

The bounds preserve exact safe integers and predictable dynamic-programming work in the browser. Larger exact rows should use arbitrary-precision storage and explicit computational limits. The bound keeps browser work predictable and every returned integer within JavaScript's exact safe range. Larger inputs need arbitrary-precision software, explicit resource controls, and independent verification rather than treating an overflowed floating-point value as exact.

Use boundary

Calculation path

Use the second-kind recurrence: place the newest labelled element into one of k existing blocks or make it the first member of a new block. The workspace preserves the bounded integer method and verification evidence beside the result.

Calculation path

S(n,k) = k S(n-1,k) + S(n-1,k-1).