Here is a problem I found on Google Plus.
Given a sequence $1,2,....,n$ you are allowed to interchange any two consecutive subsequences in it. Find the least number of steps in which you can reach $n,n-1,.....,1$ using this transformation. Consecutive subsequences means the last element of the first subsequence is less than the first element of the second subsequence.
Examples: $12345 \to 34125 \to 32541 \to 54321$ So, $T(5)=3$. Some other example are: $T(15)=11, T(13)=9, T(10)=7$
It has been found that, the transformation can always be done in at most $n-1$ ways by starting with $1,2,\ldots,n-1,n\to n,1,2...,n-1$ and then using induction. Is there a simple formula or method for finding the smallest number of ways instead of brute force?