Falling factorial ($(t{)}_j=t(t-1)\cdots (t-j+1)$)

The falling factorial is the product of $j$ successive descending factors, with $(t{)}_0=1$. Another notation is $t^{\underline{j}}$; this article uses $(t{)}_j$ for descending factors, whereas the rising factorial uses the same notation for ascending factors. For a nonnegative integer $t$, it counts ordered selections of $j$ distinct objects and vanishes when $j>t$. Powers expand as $t^j={\sum }_{\ell =0}^{j}S(j,\ell )(t{)}_{\ell }$, where $S(j,\ell )$ is a Stirling number of the second kind.