发表机构
University of Bristol（布里斯托大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证实了关于独立随机变量排序所得排列分布半代数维数的猜想，给出上界的另一证明，并回答了Ed Crane关于Mallows分布无法作为$n\geq4$时排列分布的问题
AI 中文摘要
设$X_1,\ldots,X_n$为几乎必然互异的独立实值随机变量，令$\sigma$为$\{1,\ldots,n\}$的随机排列，满足$X_{\sigma(1)}<X_{\sigma(2)}<\cdots< X_{\sigma(n)}$。我们证明，当$X_1,\ldots,X_n$的分布变化时，$\sigma$的所有分布构成的集合，作为$(n!-1)$单纯形的子集，具有半代数维数$\sum_{k=2}^n \binom{n}{k}(k-1)!$。这证实了Babson、Duchin、Iseli、Poggi-Corradini、Thurston及Tucker-Foltz的猜想，他们已证明该维数以该表达式为上界。我们通过构造$X_1,\ldots,X_n$的有限支撑分布族来提供正确维数，从而证明了该猜想。我们还利用Radford定理给出了该上界的另一证明。此外，我们证明当$n\geq4$时，排列上的Mallows分布无法作为$\sigma$的分布出现，回答了Ed Crane的问题。
英文摘要
Let $X_1,\ldots,X_n$ be almost surely distinct independent real-valued random variables. Let $σ$ be the random permutation of $\{1,\ldots,n\}$ such that $X_{σ(1)}<X_{σ(2)}<\cdots< X_{σ(n)}$. We show that the set of laws of $σ$, as the laws of $X_1,\ldots,X_n$ vary, has semialgebraic dimension \[ \sum_{k=2}^n {n \choose k}(k-1)! \] as a subset of the $(n!-1)$-simplex. This establishes a conjecture of Babson, Duchin, Iseli, Poggi-Corradini, Thurston, and Tucker-Foltz who proved that the dimension is upper bounded by the above expression. We prove their conjecture by constructing a family of finitely supported laws for $X_1,\ldots,X_n$ that provides the correct dimension. We also give an alternative proof of the upper bound using a theorem of Radford. Furthermore, we show that the Mallows law on permutations cannot arise as a law of $σ$ for $n\geq 4$, answering a question of Ed Crane.
