Is there a factorial for factorials?

Question

Is there a more succinct way to notate this?

$$(n!)((n-1)!)((n-2)!)\cdots(2!)(1!)$$

for clarification, if I had asked a similar question, how to succinctly notate:

$$(n)(n-1)(n-2)\cdots(2)(1)$$

I would be looking for this as an answer: $n!$

Answer 1

I think the most informative way would be to rewrite the product into

$$\prod_{i=1}^n i^{(n-i + 1)}$$

Answer 2

It is easy to see that yours is

$$n^1(n-1)^2(n-2)^3\cdots 3^{n-2}2^{n-1}1^{n}=\prod_{i=0}^{n-1}(n-i)^{i+1}.$$