AI 中文总结
研究给出避免(132,213)的循环排列增长率的首个非平凡下界。构建新约简过程，迭代可判断是否为循环排列，反向能构建此类排列。还得出奇偶大小类双射及受限层数排列精确计数等结果。
AI 中文摘要
我们构建了一个新的约简过程，它将一个避免(132,213)的排列转化为一个更短的排列，当且仅当原排列是循环排列时新排列才是循环排列。通过迭代此过程可确定给定的避免(132,213)的排列是否为循环排列。反向操作可得到四个步骤，能从1（n为奇数时）或21（n为偶数时）唯一地构建出每个避免(132,213)的循环排列。主要应用是给出了\(\mathcal{C}_n(132,213)\)（长度为n且避免132和213的循环排列）增长率的首个非平凡下界。还给出了约简的其他一些结果，包括奇偶大小类之间的双射以及对具有受限层数排列的精确计数。
英文摘要
We construct a new reduction process which takes a $(132,213)$-avoiding permutation to a shorter one that is cyclic if and only if the original was. Iterating it determines whether a given $(132,213)$-avoiding permutation is cyclic. Reversing it gives four moves that build every cyclic $(132,213)$-avoiding permutation, uniquely, from $1$ if $n$ is odd, and $21$ if $n$ is even. Our main application is the first non-trivial lower bound for the growth rate of $\mathcal{C}_n(132,213)$, the cyclic permutations of length $n$ avoiding $132$ and $213$. We also give several other consequences of the reduction, including a bijection between the odd and even size classes and an exact enumeration for those permutations with a restricted number of layers.
Comments17 pages
