KBB2 Problem 10

Let $P$ be a permutation of the number $(1,2,\cdots,n)$. We apply a process as follows:

Start with the leftmost number and move to the right. Everytime we encounter a number that's bigger than the leftmost number, we swap that number and the leftmost number. Continue until we arrive at the rightmost number.

Define the rank of $P$ as the number of swaps that happened. For example, the permutation$(2,3,1,4)$ has rank 2 because:

$\displaystyle (2,3,1,4) \to (3,2,1,4) \to (4,2,1,3)$

Let$S_n$ be the sum of the ranks of all possible permutations of $(1,2,\cdots,n)$. Is there an $n$ for which$S_n > 2009n!$?