在逆序列上迭代莱默码：卡塔兰不动点与有限稳定化
Iterating the Lehmer code on inversion sequences: Catalan fixed points and finite stabilization
AI总结：
本文研究有限整数序列上的算子$\boldsymbol{\theta}$，刻画其不动点、首次像为不动点的序列，证明其与卡塔兰数、Dyck路径的关联，还证明所有逆序列存在有限稳定化并给出相关性质。
AI中文摘要：
我们研究有限整数序列上的算子$\boldsymbol{\theta}$，其中$\boldsymbol{\theta}(\boldsymbol{\theta})_i$统计$\boldsymbol{\theta}_i$左侧严格小于它的元素个数，该算子是莱默码的变体。对任意序列$\boldsymbol{\theta}$，其像$\boldsymbol{\theta}(\boldsymbol{\theta})$是逆序列；$\boldsymbol{\theta}$在$[0,n-1]$的置换上的限制是到长度为$n$的逆序列的双射。我们通过避免模式$101$结合饱和条件刻画$\boldsymbol{\theta}$的不动点，证明其数量为卡塔兰数，并给出与Dyck路径的显式递归双射。还证明首次$\boldsymbol{\theta}$像为不动点的序列恰好是同时避免$101$和$201$的序列。最后，证明所有逆序列都存在有限稳定化，给出达到最大稳定时间的序列族，并证明第二稳定水平在经典模式下不封闭。
英文摘要：
We study an operator $Θ$ on finite integer sequences, where $Θ(σ)_i$ counts the entries to the left of $σ_i$ that are strictly smaller than $σ_i$. This operator is a variant of the so-called Lehmer code. For every sequence $σ$, the image $Θ(σ)$ is an inversion sequence, and the restriction of $Θ$ to permutations of $[0,n-1]$ is a bijection onto inversion sequences of length $n$. We characterize the fixed points of $Θ$ by avoidance of the pattern $101$ together with a saturation condition, prove that they are counted by the Catalan numbers, and give an explicit recursive bijection with Dyck paths. We also show that the sequences whose first $Θ$-image is fixed are precisely those avoiding both $101$ and $201$. Finally, we prove finite stabilization for all inversion sequences, exhibit a family attaining the maximal stabilization time, and show that the second stabilization level is not closed under classical patterns.
