Some experiments with complete and elementary symmetric functions
Using computer calculations the author notes some nice determinantal formulas for the classical Stirling numbers of the first and second kind, like \[ \det \left(\begin{bmatrix} i+k\cr j\end{bmatrix}\right)_{1\leq i,j\leq n}=k!^n,\qquad \det \left(\left\{\begin{matrix} i+k\cr j\end{matrix}\right\}\right)_{1\leq i,j\leq n}=n!^k, \] or \[ \det\left(\begin{bmatrix} i+k\cr j+k-1\end{bmatrix}\right)_{1\leq i,j\leq n}=\left\{\begin{matrix} n+k\cr k\end{matrix}\right\},\qquad \det \left(\left\{\begin{matrix} i+k\cr j+k-1\end{matrix}\right\}\right)_{1\leq i,j\leq n}=\begin{bmatrix} n+k\cr k\end{bmatrix}. \] He proves that these are particular cases of general formulas for complete and elementary symmetric functions.
scientific article; zbMATH DE number 6528019