Geodesic
On the number of pancake stacks requiring four flips to be sorted
Discrete mathematics & theoretical computer science, Permutation Patters 2018, Tome 21 (2019) no. 2.

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Using existing classification results for the 7- and 8-cycles in the pancake graph, we determine the number of permutations that require 4 pancake flips (prefix reversals) to be sorted. A similar characterization of the 8-cycles in the burnt pancake graph, due to the authors, is used to derive a formula for the number of signed permutations requiring 4 (burnt) pancake flips to be sorted. We furthermore provide an analogous characterization of the 9-cycles in the burnt pancake graph. Finally we present numerical evidence that polynomial formulae exist giving the number of signed permutations that require $k$ flips to be sorted, with $5\leq k\leq9$. 
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     title = {On the number of pancake stacks requiring four flips to be sorted},
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Blanco, Saúl A.; Buehrle, Charles; Patidar, Akshay. On the number of pancake stacks requiring four flips to be sorted. Discrete mathematics & theoretical computer science, Permutation Patters 2018, Tome 21 (2019) no. 2. doi : 10.23638/DMTCS-21-2-5. http://geodesic.mathdoc.fr/articles/10.23638/DMTCS-21-2-5/

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