Is there a name for an 'incomplete' factorial $\frac{n!}{m!}$? I noticed I was computing 

$${n! \over m!} ,$$

where $n > m$, inefficiently, as 
$$\frac{\prod_{k=1}^{n} k}{\prod_{k=1}^m k},$$
when many terms cancel out and I could just be calculating
$$\prod_{k=m+1}^n k .$$
Before I give this function a name, is there already a name or notation for it?  "Partial factorial"? "Incomplete factorial"?
 A: We can write a ratio of factorials as the falling factorial or descending factorial or lower factorial. We can write it using the notation
$$x^{\underline k} := x (x - 1) \cdots (x - k + 1)$$
(the Pochhammer symbol notation $(x)_k$ is also common): By cancellation we have
$$\color{#bf0000}{\boxed{\frac{n!}{m!} = n^{\,\underline {n - m}}}},$$
but of course $x^{\underline k}$ is perfectly defined for noninteger arguments $x$ too.
Anyway, the factorial notation here is surely much more familiar, and one probably couldn't use either of the other notations without comment. (Alternatively, we can write the above ratio as a rising factorial using the analogous notation $\color{#bf0000}{\smash{(m + 1)^{\overline {n - m}}}}$ or the Pochhammer notation $\color{#bf0000}{(m + 1)^{(n - m)}}$; the latter has the potential to be confusing for obvious reasons.)
A: Hint: The paper Two notes on notation by D. Knuth provides a sound reasoning about notational aspects of Iverson brackets and Stirling numbers.

D. Knuth also discusses factorial powers. He introduces in (2.11) and (2.12)

*

*$z$ to the $n$ falling: $z^{\underline{n}}=z(z-1)\cdots (z-n+1)$ and


*$z$ to the $n$ rising: $z^{\overline{n}}=z(z+1)\cdots (z+n-1)$

