An order statistics problem with some interesting geometry

Question

Let $a_n$ be a given sequence of positive numbers, and $X_n$ a sequence of independent random variables with each $X_n$ uniformly distributed on $[0, a_n]$.

Question: Let $N \geq 2$ be an arbitrary integer. What is

$$\mathbb P(X_N > X_{N-1} > \dots > X_1)$$

for the following choices of $a_n$?

i) Arithmetic progression: $a_n = n$.

ii) Geometric progression: $a_n = 2^n$.

iii) Square numbers: $a_n = n^2$.

Comments:

The problem seems to boil down to computing the (normalized) volume of some complicated $N$-dimensional shapes. For example, in the simplest case of a constant sequence $a_n = 1$, the probability is the volume of the standard $N$-simplex, $\frac{1}{N!}$.

Update: The problems (1) and (2) are effectively solved by the work of Richard Stanley, as mentioned in the comments. The case of square numbers seems quite interesting to analyze.

Answer 1

Not an answer, but a guess that may help someone... Based on computing the first few values $P_N:=\mathbb{P}(X_N>X_{N-1}>\cdots>X_1)$ it is not too difficult to guess the formulas $$P_N = \frac{(N+1)^{N-1}}{(N!)^2}$$ for $a_n=n$ and the generating function $$\sum_{N=0}^\infty P_N x^N = \frac{1}{\sum_{k=0}^\infty \frac{(-1)^k}{k!}\, 2^{-k(k-1)/2} x^k}$$ for $a_n = 2^n$.