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 $5\leq k\leq9$. Comment: We have finalized for the paper for publication in DMTCS, updated a reference to its published version, moved the abstract to its proper location, and added a thank you to the referees. The paper has 27 pages, 6 figures, and 2 tables
Discrete Mathematics & Theoretical Computer Science