Given n pancakes, for each permutation we can compute the minimum number of pancake flips. If we take the maximum over all possible permutations, we get the worst case pancake number $P_n$.
I think I can prove that $P_n \geq n$. My argument is that I can start from the sorted pancake, and do a "anti-sorting" of the pancake by first flipping at position n, then position at n-1, n-2, etc.
For example, the case of n=5 would yield 34251.
Sorting such a pancake would take at best n steps.
Am I doing something wrong?