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The Stirling2(n,m) command computes the Stirling numbers of the second kind from the well-known formula in terms of the binomial coefficients.

Instead of Stirling2 you can also use the synonym combinat[stirling2].

Regarding combinatorial functions, $\mathrm{Stirling2}\left(n\,m\right)$ is the number of ways of partitioning a set of n elements into m non-empty subsets. The Stirling numbers also enter binomial series, Mathieu function formulas, and are relevant in applications in Physics.

$\mathrm{Stirling2}\left(n\,m\right)$

$\mathrm{Stirling2}\left(n\,m\right)$

$=\mathrm{convert}\left(\,\mathrm{Sum}\right)$

$\mathrm{Stirling2}\left(n\,m\right)=\sum _{\mathrm{\_k1}=0}^m\frac{\left(\genfrac{}{}{0ex}{}{m}{\mathrm{\_k1}}\right){\mathrm{\_k1}}^n}{m!{\left(\mathrm{-1}\right)}^{-m+\mathrm{\_k1}}}$

$\mathrm{eval}\left(\,\left[n=10\,m=5\right]\right)$

$42525=\sum _{\mathrm{\_k1}=0}^5\frac{\left(\genfrac{}{}{0ex}{}{5}{\mathrm{\_k1}}\right){\mathrm{\_k1}}^{10}}{120{\left(\mathrm{-1}\right)}^{-5+\mathrm{\_k1}}}$

$\mathrm{value}\left(\right)$

$42525=42525$