Given $n$ numbers. We permute these numbers uniformly at random and arrange them on circle. What is the expected number of local maxima?
Note: We call a local maximum a number that is greater than both of its neighbors.
This problem is similar to this one except that here we are dealing with numbers arranged on the circle. Does $(n + 1)/3$ hold for this problem?
It is close. The $+1$ comes from the endpoints, which are more likely to be maxima as they only have one neighbor. The linked answer shows the chance for an interior point to be a local maximum is $\frac 13$, so by the linearity of expectation the expected total number is $\frac n3$.