排列中匹配的极值族
Extremal Families for Matchings in Permutations
- Institute for Mathematical Sciences, Renmin University of China（中国人民大学数学科学研究所）
- Department of Mathematical Sciences, Tsinghua University（清华大学数学科学系）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文在2≤s≤n的全范围刻画了排列族不含s个两两不相交排列时的极值等号情形，证明该情形等价于族为(s-1)个两两不相交1-陪集的并集，并给出相关线性命题的组合证明。
AI中文摘要：
两个排列σ,τ∈Sₙ被称为不相交的，若复合映射στ⁻¹没有不动点。若族F⊆Sₙ不含s个两两不相交的排列，通过简单平均论证可得|F|≤(s-1)(n-1)!。Inozemtsev、Kolupaev与Kupavskii在s≤n/(2¹⁷logn)范围内刻画了等号成立的情形。本文在2≤s≤n的整个范围内刻画了所有等号情形：等号成立当且仅当F是(s-1)个两两不相交的1-陪集的并集。本文还证明了该分类背后的线性命题：Sₙ上的实值函数在每个1-因子分解上的和为常数，当且仅当该函数属于1-陪集指示函数的张成空间。证明为组合性的，适用于所有阶数，少数小阶数单独处理。
英文摘要：
Two permutations $σ,τ\in S_n$ are called disjoint if the composition $στ^{-1}$ has no fixed point. If a family $\mathcal F\subseteq S_n$ contains no $s$ pairwise disjoint permutations, then a simple averaging argument gives $|\mathcal F|\leq(s-1)(n-1)!$. Inozemtsev, Kolupaev and Kupavskii characterized the equality cases in the range $s\leq n/(2^{17}\log n).$ We characterize all equality cases throughout the range $2\le s\le n$: equality holds if and only if $\mathcal F$ is a union of $(s-1)$ pairwise disjoint $1$-cosets. We also prove the linear statement underlying this classification: a real-valued function on $S_n$ has constant sum on every one-factorization if and only if it lies in the span of the indicators of the $1$-cosets. The proof is combinatorial and applies to every order, with a few small orders handled separately.
