Particular Values of Stirling Numbers of the Second Kind

Theorem

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

Stirling Number of the Second Kind: $\ds {0 \brace n}$

$\ds {0 \brace n} = \delta_{0 n}$

Stirling Number of the Second Kind: $\ds {1 \brace n}$

$\ds {1 \brace n} = \delta_{1 n}$

Stirling Number of the Second Kind: $\ds {n \brace n}$

$\ds {n \brace n} = 1$

Stirling Number of the Second Kind: $\ds {n \brace n - 1}$

$\ds {n \brace n - 1} = \binom n 2$

Stirling Number of the Second Kind: $\ds {n \brace n - 2}$

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

Stirling Number of the Second Kind: $\ds {n \brace n - 3}$

$\ds {n \brace n - 3} = \binom {n + 2} 6 + 8 \binom {n + 1} 6 + 6 \binom n 6$

Stirling Number of the Second Kind: $\ds {n + 1 \brace 0}$

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

Stirling Number of the Second Kind: $\ds {n + 1 \brace 1}$

$\ds {n + 1 \brace 1} = 1$

Stirling Number of the Second Kind: $\ds {n + 1 \brace 2}$

$\ds {n + 1 \brace 2} = 2^n - 1$

Also see

• Particular Values of Unsigned Stirling Numbers of the First Kind
• Particular Values of Signed Stirling Numbers of the First Kind