On the number of pancake stacks requiring 4 flips to be sorted
share
Using an 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 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 permutations that require k flips to be sorted, with k=5,6. We present similar conjectures for sign permutations and for 5≤ k≤9.
READ FULL TEXT