Stirling Number Second Kind Calculator

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.

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.

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.