There are $n$ sticks of lengths $1$ to $n$ in a row. Upon permuting them randomly, we may calculate the average number of peaks viewed from left. A peak is a stick such that all sticks to its left are shorter than it.
For example, suppose there are $3$ sticks. The permutations and numbers of peaks are:
123 3132 2213 2231 2312 1321 1The total number of peaks is $3+2+2+2+1+1=11$ and there are $6$ permutations, so the average number of peaks is $11/6\approx1.83333$.
The question is: as $n\to\infty$, how does the average number of peaks grow?
I have written a program to calculate the results for small $n$'s, with output shown below:
1 12 1.53 1.833334 2.083335 2.283336 2.457 2.592868 2.717869 2.8289710 2.9289711 3.0198812 3.1032113 3.18013It looks like that the growth is $\sqrt n$ or $\log n$, but I don't know which, do you?
