关于需要四次翻转才能排序的煎饼堆数量
On the number of pancake stacks requiring four flips to be sorted
AI总结：
本文利用煎饼图及烧焦煎饼图中环的分类结果，确定了需要4次翻转才能排序的排列数量，并推导了符号排列的公式，同时给出5至9次翻转情形下存在多项式公式的数值证据。
摘要：
利用煎饼图中7-环和8-环的现有分类结果，我们确定了需要4次煎饼翻转（前缀反转）才能排序的排列数量。作者先前对烧焦煎饼图中8-环的类似刻画，被用来推导需要4次（烧焦）煎饼翻转才能排序的符号排列数量的公式。我们进一步提供了烧焦煎饼图中9-环的类似刻画。最后，我们给出数值证据，表明存在多项式公式，给出需要$k$次翻转才能排序的符号排列数量，其中$5\leq k\leq9$。
英文摘要：
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 $5\leq k\leq9$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
