What is $\frac{(an)!}{n!}$?

Question

How can $\frac{(an)!}{n!}$ be expressed in terms of $a!$, $n!$, $a$, $n$ (and maybe Pochhammers)?

Answer 1

If you don't insist on being exact, you can use Stirling's approximation. $$\frac{(an)!}{n!}\approx \frac {(an)^{an}e^n}{n^ne^{an}}\sqrt{a}=a^{an}n^{an-1}e^{-(an-1)}\sqrt a$$ which will be very close-within about a factor of $1+\frac 1{12n}$

Answer 2

By OP's request, my above comment was that since $m!=(m)_m$, we may write the desired expression as $$\frac{(an)_{an}}{a!}$$

Note: this solution is not the one @lhf commented on, below.

Answer 3

For $n>m$, $$ \frac{n!}{m!} = P(n,n-m) = \text{the number of permutations of $n-m$ in a set of $n$}. $$ This is one context in which it doesn't make sense to define "permutation" as a bijection from a set to itself, but rather one should define it as an ordered list of length $n-m$ of members of a set of size $n$.