Particular Values of Stirling Numbers of the Second Kind

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Theorem

This page gathers together some particular values of Stirling numbers of the second kind.


Stirling Number of the Second Kind: $\displaystyle \left\{ {0 \atop n}\right\}$

$\displaystyle \left\{ {0 \atop n}\right\} = \delta_{0 n}$


Stirling Number of the Second Kind: $\displaystyle \left\{ {1 \atop n}\right\}$

$\displaystyle \left\{ {1 \atop n}\right\} = \delta_{1 n}$


Stirling Number of the Second Kind: $\displaystyle \left\{ {n \atop n}\right\}$

$\displaystyle \left\{ {n \atop n}\right\} = 1$


Stirling Number of the Second Kind: $\displaystyle \left\{ {n \atop n - 1}\right\}$

$\displaystyle \left\{ {n \atop n - 1}\right\} = \binom n 2$


Stirling Number of the Second Kind: $\displaystyle \left\{ {n \atop n - 2}\right\}$

$\displaystyle {n \brace n - 2} = \binom {n + 1} 4 + 2 \binom n 4$


Stirling Number of the Second Kind: $\displaystyle \left\{ {n \atop n - 3}\right\}$

$\displaystyle \left\{ {n \atop n - 3}\right\} = \binom {n + 2} 6 + 8 \binom {n + 1} 6 + 6 \binom n 6$


Stirling Number of the Second Kind: $\displaystyle \left\{ {n + 1 \atop 0}\right\}$

$\displaystyle {n + 1 \brace 0} = 0$


Stirling Number of the Second Kind: $\displaystyle \left\{ {n + 1 \atop 1}\right\}$

$\displaystyle \left\{ {n + 1 \atop 1}\right\} = 1$


Stirling Number of the Second Kind: $\displaystyle \left\{ {n + 1 \atop 2}\right\}$

$\displaystyle \left\{ {n + 1 \atop 2}\right\} = 2^n - 1$


Also see