Question 127
Define the annoyingness of a permutation of the first $n$ integers to be the minimum number of copies of the permutation that are needed to be placed next to each other so that the subsequence $1,2, \ldots, n$ appears. For instance, the annoyingness of $3,2,1$ is 3, and the annoyingness of $1,3,4,2$ is 2.
A random permutation of $1,2, \ldots, 2022$ is selected. Compute the expected value of the annoyingness of this permutation.
A random permutation of $1,2, \ldots, 2022$ is selected. Compute the expected value of the annoyingness of this permutation.