Is there a name for $\frac{n!}{m!}$? Is there a name or short way of writing of $\frac{n!}{m!}$? I've searched and the closest I could find was binomial coefficient. Is there any other way?

That tends to be called a _falling_ or _rising_ factorial, and there are multiple notations, though none of them are standard enough that you could use one in a paper without defining it first.