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 $5\leq k\leq9$.

Source : oai:arXiv.org:1902.04055

Volume: Vol. 21 no. 2, Permutation Patters 2018

Published on: November 4, 2019

Submitted on: February 27, 2019

Keywords: Computer Science - Discrete Mathematics,Mathematics - Combinatorics,05A15, 05A05, 68R10,G.2.0,G.2.1,G.2.2