排列及其旋转中的模式避免
Pattern avoidance in permutations and their rotations
中文总结 AI 辅助
研究排列及其旋转中的模式避免问题，给出前\(k\geq2\)次旋转都避免给定长度为三的模式的排列数等精确公式，所研究的威尔夫等价类由补和反转决定，还对前两次旋转避免不同模式的排列进行分类枚举。
中文摘要 AI 辅助
排列的旋转是通过将排列的前几项移到排列末尾得到的新排列。循环排列是一个排列的所有旋转的集合。长度为三、四的排列和循环排列的模式避免的枚举已得到充分研究。本文给出了前\(k\geq2\)次旋转都避免给定长度为三的模式的排列数，以及前三次旋转分别避免给定长度为三的模式的旋转的排列数的精确公式。与避免长度为三的模式的排列和循环排列不同，所研究的威尔夫等价类完全由补和反转决定。我们还对前两次旋转避免不同长度为三的模式的排列进行了分类和枚举。
英文摘要
A rotation of a permutation is a new permutation obtained by moving the first several terms of the permutation to the end of the permutation. A circular permutation is the set of all rotations of a permutation. The enumerations of permutations and circular permutations avoiding patterns of length three and four are well studied. In this paper, we provide exact formulas for the number of permutations whose first $k\geq 2$ rotations all avoid a given pattern of length three, as well as the number of permutations whose first three rotations respectively avoid the rotations of a given pattern of length three. In contrast to permutations and circular permutations avoiding patterns of length three, the Wilf-equivalence classes under study are entirely determined by complements and reverses. We also classify and enumerate permutations whose first two rotations avoid different patterns of length three.
