Stirling Number of the Second Kind of 1

Theorem

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

where:

$\ds {1 \brace n}$ denotes a Stirling number of the second kind

$\delta_{1 n}$ is the Kronecker delta.

Proof

\(\ds {1 \brace n}\)

\(=\)

\(\ds n \times {0 \brace n} + {0 \brace n - 1}\)

Definition of Stirling Numbers of the Second Kind

\(\ds \)

\(=\)

\(\ds n \times \delta_{0 n} + \delta_{0 \paren {n - 1} }\)

Definition of Stirling Numbers of the Second Kind

\(\ds \)

\(=\)

\(\ds \delta_{0 \paren {n - 1} }\)

as $n \times \delta_{0 n}$ is either $n \times 0$ or $0 \times 1$ and so always $0$

\(\ds \)

\(=\)

\(\ds \delta_{1 n}\)

$0 = n - 1 \iff 1 = n$

$\blacksquare$

Also see

• Unsigned Stirling Number of the First Kind of 1
• Signed Stirling Number of the First Kind of 1

• Particular Values of Stirling Numbers of the Second Kind