Duality Law for Stirling Numbers

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Theorem

For all integers $n, m \in \Z$:

$\ds {n \brace m} = {-m \brack -n}$

where:

$\ds {n \brace m}$ denotes a Stirling number of the second kind
$\ds {n \brack m}$ denotes an unsigned Stirling number of the first kind.


Proof


Sources