Exact number of flips required to sort a burnt stack of pancakes

In this work, we consider the burnt pancake problem, which is a well-studied problem going back to a work of Gates and Papadimitriou from 1979.The problem is to sort a stack of~$n$ one-sided burnt pancakes of different sizes, by a sequence of flips of the top pancakes, such that at the end of the flipping sequence the pancakes have increasing size and the burnt sides of all pancakes are face-down. The pancakes are denoted by $ 1,2,\dots,n$, and a number is multiplied by $-1$, if the corresponding pancake has burnt side face-up. Let $T(n)$ be the minimum number of flips to sort a special stack of $n$ pancakes, namely $\overline{I_n} := [-1,-2,...,-n]$. The instance $\overline{I_n} $ has strong relevance because of its easy structure and as it has been shown to be a worst-case instance for several small $n$. Heydari and Sudborough gave in 1997 the currently best upper bound of $T(n)$, namely $ \lfloor (3n+3)/2 \rfloor $ for $ n \equiv 3 \bmod 4$, which later has been shown to be exact by a work of Cibulka from 2011. Except these two works, no progress regarding lower and upper bounds has been made until now. In our work, we present that $ \lfloor (3n+3)/2 \rfloor $ is also an upper bound of $T(n)$ for $n \equiv 1 \bmod 4 $, which again matches the lower bound of Cibulka in 2011 and thus is exact. Furthermore, we show that our construction approach for $n \equiv 1 \bmod 4 $ and the one of Heydari and Sudborough for $n \equiv 3 \bmod 4 $ cannot be applied for even $n$. However, as there might be different construction approaches, the case of even $n$ remains an open problem, where two possible values for $T(n)$ are possible, namely $ (3/2)n + 1 $ or $ (3/2)n + 2 $. Finally, we found two values, namely $n=24$, $n=26$, where the lower bound is attained.