Problem Archive

Consider the following algorithm for sorting a list:

• 1. Starting from the beginning of the list, check each pair of adjacent elements in turn.
• 2. If the elements are out of order:
  • a. Move the smallest element of the pair at the beginning of the list.
  • b. Restart the process from step 1.
• 3. If all pairs are in order, stop.

For example, the list $\{\,4\,1\,3\,2\,\}$ is sorted as follows:

• $\underline{4\,1}\,3\,2$ ($4$ and $1$ are out of order so move $1$ to the front of the list)
• $1\,\underline{4\,3}\,2$ ($4$ and $3$ are out of order so move $3$ to the front of the list)
• $\underline{3\,1}\,4\,2$ ($3$ and $1$ are out of order so move $1$ to the front of the list)
• $1\,3\,\underline{4\,2}$ ($4$ and $2$ are out of order so move $2$ to the front of the list)
• $\underline{2\,1}\,3\,4$ ($2$ and $1$ are out of order so move $1$ to the front of the list)
• $1\,2\,3\,4$ (The list is now sorted)

Let $F(L)$ be the number of times step 2a is executed to sort list $L$. For example, $F(\{\,4\,1\,3\,2\,\}) = 5$.

Let $E(n)$ be the expected value of $F(P)$ over all permutations $P$ of the integers $\{1, 2, \dots, n\}$.
You are given $E(4) = 3.25$ and $E(10) = 115.725$.

Find $E(30)$. Give your answer rounded to two digits after the decimal point.