# 2022-15 A determinant of Stirling numbers of second kind

Let $$S(n,k)$$ be the Stirling number of the second kind that is the number of ways to partition a set of $$n$$ objects into $$k$$ non-empty subsets. Prove the following equality $\det\left( \begin{matrix} S(m+1,1) & S(m+1,2) & \cdots & S(m+1,n) \\ S(m+2,1) & S(m+2,2) & \cdots & S(m+2,n) \\ \cdots & \cdots & \cdots & \cdots \\ S(m+n,1) & S(m+n,2) & \cdots & S(m+n,n) \end{matrix} \right) = (n!)^m$

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