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"?
On
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: $z^{\overline{n}}=z(z+1)\cdots (z+n-1)$
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.)