Stirling Number of n with n-m is Polynomial in n of Degree 2m

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Theorem

Unsigned Stirling Number of the First Kind

Let $m \in \Z_{\ge 0}$.

The unsigned Stirling number of the first kind $\displaystyle \left[{n \atop n - m}\right]$ is a polynomial in $n$ of degree $2 m$.


Stirling Number of the Second Kind

Let $m \in \Z_{\ge 0}$.

The Stirling number of the second kind $\displaystyle \left\{ {n \atop n - m}\right\}$ is a polynomial in $n$ of degree $2 m$.


Sources