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

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.)