Or stay pit to be filled.
\ (\ begin {bmatrix} n \\ m \ end {bmatrix} \) represents the number of a first type Stirling
\ (\ begin {Bmatrix} n \\ m \ end {Bmatrix} \) represents the number of the second type Stirling
The first number Stirling
Gugu Gu
Stirling number of second kind
definition:
\ (\ begin {Bmatrix} n \\ m \ end {Bmatrix} \) shows a \ (n-\) th unlabeled beads into \ (m \) number of unlabeled box scheme
Recursive:
\[\begin {Bmatrix} n \\ m\end {Bmatrix} = \begin {Bmatrix} n-1 \\ m-1\end {Bmatrix}+m*\begin {Bmatrix} n-1 \\ m\end {Bmatrix}\]
nature:
\[n^k=\sum \limits _{i=0}^k \begin {Bmatrix} k \\ i\end {Bmatrix}*n^{\underline k}=\sum \limits _{i=0}^k \begin {Bmatrix} k \\ i\end {Bmatrix}*\binom{n}{i}*i!\]
Muttered to prove