Stirling numbers of the first kind

Definition

Let $n$ and $k$ be natural numbers. The unsigned Stirling number of the first kind $c(n, k)$ is the number of permutations of an $n$-element set that have exactly $k$ disjoint cycles. Fixed points count as cycles of length $1$.

Remarks

Other notations are $\left[ {n \atop k} \right]$ and $|s(n, k)|$. The signed numbers are $s(n, k) = (-1)^{n-k} c(n, k)$. Example: $c(3, 2) = 3$. The permutations of $\{1, 2, 3\}$ with two cycles are the three transpositions. The numbers satisfy $\sum_k c(n, k) = n!$ and $x (x + 1) \cdots (x + n - 1) = \sum_k c(n, k)\, x^k$. The values satisfy a recurrence in $n$ and $k$.

Explanations

L1 human

Write a permutation as a set of cycles, for example $(1\,2)(3)$. $c(n, k)$ counts the permutations of $n$ elements that split into exactly $k$ cycles.