For questions about the two kinds of Stirling numbers and related topics, such as Lah numbers.
There are two kinds of Stirling numbers:
Stirling numbers of the first kind ${n\brack k}$ count the number of ways to permute $n$ objects into $k$ cycles.
Stirling numbers of the second kind ${n\brace k}$ count the number of ways to arrange $n$ objects into $k$ non-empty subsets.
Each sequence satisfies a recurrence:
\begin{align*} {n+1 \brack k} &= n {n \brack k} + {n \brack k-1} \\ {n+1 \brace k} &= k {n \brace k} + {n \brace k-1} \end{align*}
Lah numbers are closely related, being $$L(n,k) = \sum_{j=0}^n {n \brack j}{j \brace k}$$
