I'm looking for a citable reference for the following identity involving the Stirling numbers of the second kind $S(n, k)$ stated in [Equation (27)](https://mathworld.wolfram.com/StirlingNumberoftheSecondKind.html): For $n \geq 2$,
$$
\sum\_{m=1}^n S(n, m) (-1)^m (m-1)! = 0.
$$
Thank you.



Identity involving Stirling number of the second kind
The first formula in Section 24.1.4.I.B in Handbook of mathematical functions with formulas, graphs, and mathematical tables, Dover edition, 1965 (Library of Congress Catalog Card Number: 65.12253) by Abramowitz and Stegun is
$$
x^n=\sum\_{m=0}^n S(n,m) x(x-1)\cdots(x-m+1). \tag{1}\label{1}
$$
Assuming here $n>1$ (so that $S(n,0)=0$), dividing both sides by $x$, and then letting $x=0$, we get the desired identity
$$\sum\_{m=1}^n S(n,m)(-1)^m(m-1)!=0.$$




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One may note that Riordan (An Introduction to Combinatorial Analysis, Wiley, 1958, formula (35) on p. 33) uses identity \eqref{1} to *define* the Stirling numbers $S(n,m)$, then deriving their combinatorial definition (as the number of ways of partitioning a set of $n$ elements into $m$ nonempty sets) -- cf. formula (38) on p. 33 and the third display from the bottom on p. 91 of the book.


