Duality Law for Stirling Numbers

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Theorem

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

$\displaystyle \left\{ {n \atop m}\right\} = \left[{-m \atop -n}\right]$

where:

$\displaystyle \left\{ {n \atop m}\right\}$ denotes a Stirling number of the second kind
$\displaystyle \left[{n \atop m}\right]$ denotes an unsigned Stirling number of the first kind.


Proof


Sources