Difference between revisions of "Stirling numbers of the second kind"
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The Stirling numbers of the second kind, commonly written as $S(n,k)$ or $\begin{Bmatrix}n \\ k\end{Bmatrix}$ are given by | The Stirling numbers of the second kind, commonly written as $S(n,k)$ or $\begin{Bmatrix}n \\ k\end{Bmatrix}$ are given by | ||
$$S(n,k)=\dfrac{1}{k!} \displaystyle\sum_{j=0}^{k} (-1)^{k-j} {k \choose j} j^n,$$ | $$S(n,k)=\dfrac{1}{k!} \displaystyle\sum_{j=0}^{k} (-1)^{k-j} {k \choose j} j^n,$$ | ||
| − | where ${k \choose j}$ denotes a [[binomial coefficient]]. The Stirling numbers of the second kind appear in the definition of the [[Bell numbers]]. | + | where ${k \choose j}$ denotes a [[binomial coefficient]]. The Stirling numbers of the second kind appear in the definition of the [[Bell numbers]] and [[Touchard polynomial|Touchard polynomials]]. |
Revision as of 04:29, 27 June 2015
The Stirling numbers of the second kind, commonly written as $S(n,k)$ or $\begin{Bmatrix}n \\ k\end{Bmatrix}$ are given by $$S(n,k)=\dfrac{1}{k!} \displaystyle\sum_{j=0}^{k} (-1)^{k-j} {k \choose j} j^n,$$ where ${k \choose j}$ denotes a binomial coefficient. The Stirling numbers of the second kind appear in the definition of the Bell numbers and Touchard polynomials.
