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

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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.

Mar 14, 2022

#1
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I did that at length.

I am still waiting on a response from you.

Mar 14, 2022
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Have responded. As I said, I will work on characterizing all numbers that serve as counter-examples.

I'm curious to know an approach to this post too.

#3
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Thanks, :)

Unfortunately I do not think this question will get my attention.

It looks very time consuming and it doesn't look like my kind of question.