Stirling Number of the First Kind Calculator

How it is calculated

The unsigned Stirling number of the first kind c(n, k) counts the permutations of n items that split into exactly k cycles. The tool builds a table with the recurrence c(n, k) = c(n−1, k−1) + (n−1) × c(n−1, k), starting from c(0, 0) = 1. The idea: item n is either alone in its own cycle, which gives the first term, or it is inserted after any of the other n−1 items in an existing cycle, which gives the second.

Inputs are rounded to whole numbers, k must not exceed n, both must be zero or more, and n is limited to 20; anything else gives no result. For n above 0, c(n, 0) is 0, and c(0, 0) is 1. The results are exact integers, even at n = 20 where the numbers have 18 digits, and they are grouped with your language's thousands separator.

Common mistakes

• Confusing the first kind with the second kind. The first kind counts permutations by cycles; the second kind counts ways to split a set into groups, and the numbers differ (for n = 6, k = 3 the second kind gives 90).
• Expecting signed values. This tool gives the unsigned (absolute) numbers; the signed version alternates signs and would show negative numbers.
• Entering k larger than n. There is no way to make more cycles than items, so the tool returns nothing rather than 0.
• Entering n above 20. The tool stops at 20, and n = 21 gives no result.
• Typing a decimal such as 5.6 and expecting an error. It is rounded to 6 first, so the result shown belongs to n = 6, and the echoed n confirms which value was used.