Stirling Numbers Calculator

How it is calculated

Both numbers are built from small cases with exact integer arithmetic. Second kind, the ways to split n items into k non-empty groups: S(n,k) = k·S(n−1,k) + S(n−1,k−1). First kind, the permutations of n items with exactly k cycles: c(n,k) = (n−1)·c(n−1,k) + c(n−1,k−1). Both start from S(0,0) = c(0,0) = 1.

The first-kind value is the unsigned one; the signed Stirling number of the first kind has the sign (−1)^n−k. The inputs n and k are rounded to whole numbers and must be between 0 and 25; values outside that range return no result. If k is larger than n, or k is 0 with n above 0, both results are 0. Big values are printed in full with thousands separators, for example n = 25 and k = 10 gives numbers with 19 and 22 digits.

Common mistakes

• Mixing up the two kinds: the second-kind value (15 here) counts set partitions, while the first-kind value (50) counts permutations by cycles. They match only in trivial cases such as k = n, where both are 1.
• Expecting the signed first-kind number: the tool returns the unsigned count, so no minus sign appears even when n − k is odd.
• Treating groups as ordered: partitions counted by S(n,k) are unordered, so splitting into {1,2} and {3,4,5} is the same as the reverse order. Multiply by k! to count ordered, onto assignments.
• Using n above 25: the tool stops at 25 to keep every digit exact, so larger values return no result rather than a rounded number.
• Entering k = 0 for n above 0 and expecting 1: there is no way to place items into zero non-empty groups, so both results are 0. Only n = 0 and k = 0 gives 1.