How does this turn into... Harmonic Numbers?!

Define the following statistic as follows:

$D(a_1,a_2,\ldots, a_n)$:

Start at $a_n$. Go back from $a_n$ until you get to the nearest element of the list smaller than it. Suppose that the first element smaller than it is $a_k$. We break the list there (right after $a_k$).

Then, repeat from $a_k$... go back until you get nearest element in the list smaller than $a_k$.

By the end, you will have split the list a certain number of times, x, creating x+1 sub-lists.

Now, consider the expression:

G(n) = $\frac{D(1,2,3, \ldots , (n-1) ,n) + D(1,2,3, \ldots , n ,( n-1) , \ldots + D(n , (n-1) , \ldots , 2 , 1)}{n!}$ where the numerator is $D$ evaluated at all $n!$  permutations of $1,2,3,4,\ldots,n$.

For example, for n=3, we have:

$G(3) = \frac{D(1,2,3)+D(1,3,2)+D(2,1,3)+D(2,3,1)+D(3,1,2)+D(3,2,1)}{3!}$, which can be shown to equal $\frac{11}{6}.$

Prove that $G(n)$ is the $n$th harmonic number!!!!!!!!!!!!!!!!!

Thank you so much in advance.