arXiv 1402.06682014-02-04math.CO

具有固定圈数的置换的Erd{ \H o}s-Ko-Rado定理

An Erd{\H o}s-Ko-Rado theorem for permutations with fixed number of cycles

Cheng Yeaw Ku, Kok Bin Wong

备注：: 8 pages

摘要：

令$S_{n}$表示$[n]=\{1,2,\dots, n\}$的置换集。对于正整数$k$，将$S_{n,k}$定义为具有恰好$k$不相交循环的$[n]$的所有排列的集合，即，\[ S_{n,k} = \{π\in S_{n}: π= c_{1}c_{2} \cdots c_{k}\},\]，其中$c_1,c_2,\dots ,c_k$为不相交循环。$S_{n,k}$的大小由$\left [ \begin{matrix}n\\ k \end{matrix}\right]=(-1)^{n-k}s(n,k)$给出，其中$s(n,k)$是第一类斯特林数。如果$\mathcal{A}$的任何两个元素至少有$t$公共圈，则称族$\mathcal{A} \subseteq S_{n,k}$为$t$ -{ \em相交}。本文证明了，给定任意正整数$k,t$，$k\geq t+1$，存在一个整数$n_0=n_0(k,t)$，使得对所有的$n\geq n_0$，如果$\mathcal{A} \subseteq S_{n,k}$是$t$ -相交的，则\[ |\mathcal{A}| \le \left [ \begin{matrix}n-t\\ k-t \end{matrix}\right],\]具有相等性当且仅当$\mathcal{A}$是$t$不动点的稳定子.

英文摘要：

Let $S_{n}$ denote the set of permutations of $[n]=\{1,2,\dots, n\}$. For a positive integer $k$, define $S_{n,k}$ to be the set of all permutations of $[n]$ with exactly $k$ disjoint cycles, i.e., \[ S_{n,k} = \{π\in S_{n}: π= c_{1}c_{2} \cdots c_{k}\},\] where $c_1,c_2,\dots ,c_k$ are disjoint cycles. The size of $S_{n,k}$ is given by $\left [ \begin{matrix}n\\ k \end{matrix}\right]=(-1)^{n-k}s(n,k)$, where $s(n,k)$ is the Stirling number of the first kind. A family $\mathcal{A} \subseteq S_{n,k}$ is said to be $t$-{\em intersecting} if any two elements of $\mathcal{A}$ have at least $t$ common cycles. In this paper, we show that, given any positive integers $k,t$ with $k\geq t+1$, there exists an integer $n_0=n_0(k,t)$, such that for all $n\geq n_0$, if $\mathcal{A} \subseteq S_{n,k}$ is $t$-intersecting, then \[ |\mathcal{A}| \le \left [ \begin{matrix}n-t\\ k-t \end{matrix}\right],\] with equality if and only if $\mathcal{A}$ is the stabiliser of $t$ fixed points.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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