Partitionen/Stirling-Zahlen zweiter Art/Summendarstellungen/Fakt

Für die Stirling-Zahlen zweiter Art gelten die folgenden Beschreibungen.

${}S({n},{k})={\frac {1}{k!}}\sum _{(r_{1},\ldots ,r_{k}):\,r_{1}+r_{2}+\cdots +r_{k}=n,\,r_{j}\geq 1}{\binom {n}{r_{1},\dotsc ,r_{k}}}\,.$

${}S({n},{k})=\sum _{c_{1}+c_{2}+\cdots +c_{k}=n-k}1^{c_{1}}2^{c_{2}}\cdots k^{c_{k}}\,.$

${}S({n},{k})=\sum _{1\leq i_{1}\leq i_{2}\leq \ldots \leq i_{n-k}\leq k}i_{1}i_{2}\cdots i_{n-k}\,.$

${}S({n},{k})={\frac {1}{k!}}\sum _{j=0}^{k}(-1)^{k-j}{\binom {k}{j}}j^{n}\,.$

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