Lower bound on worst case pancake number? 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?
I assume that you are talking about the Pancake sorting (wiki) problem. Your argument for $P_n \ge n$ is rather informal.

The paper "Bounds for Sorting by Prefix Reversal (1979)" gives the lower bound $P_n \ge n$ in the Introduction section. Better lower bounds are also mentioned in this paper.

This paper "Pancake Flipping Is Hard" (2011) proves that determining the minimal number of moves for a given stack is NP-hard.