【读书笔记】排列研究-随机排列

目录

• 开胃菜
  • Example1
  • Example2
• Standard Young Tableaux SYT
  • 定义
  • 例子
  • 入门材料
• The Hooklength Formula
  • 先对Young表中的hook定义
  • 定理Hooklength Formula
  • 举例
• The Frobenius formula
  • 定理和例子
• 期望
• 方差标准差
• 应用-最长上升子序列

研究的是随机排列\(p\)具有性质\(A\)的概率

说实话，看到标题我首先想到的是那个\(1/e\)策略选伴侣/雇员的问题。。。

如果\(n!\)个排列都是等可能取出的话，那么不过还是计数问题，算概率再除以\(n!\)

实际中，可能会用到一些概率理论的技巧来方便解决问题

开胃菜

Example1

Let \(i\) and \(j\) be two distinct elements of \([n]\), and let \(p\) be a randomly chosen \(n\)-permutation. Then the probability that \(i\) and \(j\) belong to the same cycle of \(p\) is \(\frac{1}{2}\)

Example2

Let \(i, k \in[n],\) and let \(p\) be a randomly selected \(n\) -permutation. Then the probability that the entry \(i\) is part of a \(k\) -cycle of \(p\) is \(\frac{1}{n} ;\) in particular, it is independent of \(k .\)

Standard Young Tableaux SYT

定义

A Standard Young Tableau is a Ferrers shape on \(n\) boxes in which each box contains one of the elements of [n] so that all boxes contain different numbers, and the rows and columns increase going down and going to the right.

例子

入门材料

ams.org/notices/200702/whatis-yong.pdf

The Hooklength Formula

先对Young表中的hook定义

定义Young表中一个格子\(b\)的hook是右边和下边格子的集合(包括\(b\))，记作\(H_b\)

The size of $H_b $is called the hooklength of \(b\), and is denoted by \(h_b\).

the box \(b\) is called the peak of \(H_b\).

定理Hooklength Formula

Let \(F\) be any Ferrers shape on \(n\) boxes. Then the number of SYT of shape \(F\) is equal to

\[\frac{n !}{\prod_{b} h_{b}} \]

where the product is over all n boxes b of \(F\).

举例

The Frobenius formula

定理和例子

期望

方差标准差

应用-最长上升子序列

先空着

资料来自网络

书用的是Combinatorics of permutations by Miklos Bona