【读书笔记】排列研究-模式避免-基础Pattern Avoidance

目录

• 模式避免的定义
• 避免Pattern q 的n-排列计数\(S_n(q)\)
  • q长度是2
  • q长度是3
  • 对一些模式q,做\(S_n(q)\)的阶估计
  • Backelin, West, and Xin给出的较为一般的Theorem
  • q长度是4
• 证明Stanley-Wilf conjecture
  • The Stanley-Wilf conjecture
  • The Füredi-Hajnal conjecture

模式避免的定义

避免Pattern q 的n-排列计数\(S_n(q)\)

先扔结论，有时间把证明粘过来

q长度是2

\[S_n(12)=S_n(21)=1
\]

q长度是3

All patterns of length three are avoided by the same number of n-permutations.

\[S_n(123)=S_n(132)=S_n(213)=S_n(231)=S_n(312)=S_n(321)\\
S_{n}(132)=C_{n}=\frac{\left(\begin{array}{c}
2 n \\
n
\end{array}\right)}{n+1}
\]

对一些模式q,做\(S_n(q)\)的阶估计

Backelin, West, and Xin给出的较为一般的Theorem

举例，

\(r=2,k=2,q=34\) it says, \(S_n(1234)=S_n(2134)\)

\(r=2,k=2,q=43\) it says, \(S_n(1243)=S_n(2143)\)

\(r=3,k=1,q=4\) it says, \(S_n(1234)=S_n(3214)\)

q长度是4

本来\(q\)有24种，可以证明最后归结为这三种代表,1234,1342,1324

\[\begin{aligned}
&\text { for } S_{n}(1342) \text { we have } 1,2,6,23,103,512,2740,15485\\
&\text { for } S_{n}(1234) \text { we have } 1,2,6,23,103,513,2761,15767\\
&\text { for } S_{n}(1324) \text { we have } 1,2,6,23,103,513,2762,15793
\end{aligned}
\]

aovid 1342 A022558

avoid 1234 A005802

avoid 1324 A061552

\[\begin{aligned}
S_{n}(1342) &=(-1)^{n-1} \cdot \frac{\left(7 n^{2}-3 n-2\right)}{2} \\
&+3 \sum_{i=2}^{n}(-1)^{n-i} \cdot 2^{i+1} \cdot \frac{(2 i-4) !}{i !(i-2) !} \cdot\left(\begin{array}{c}
n-i+2 \\
2
\end{array}\right)
\end{aligned}
\]

\[S_{n}(1234)=2 \cdot \sum_{k=0}^{n}\left(\begin{array}{c}
2 k \\
k
\end{array}\right)\left(\begin{array}{l}
n \\
k
\end{array}\right)^{2} \frac{3 k^{2}+2 k+1-n-2 n k}{(k+1)^{2}(k+2)(n-k+1)}
\]

\[S_{n}(1234)=\frac{1}{(n+1)^{2}(n+2)} \sum_{k=0}^{n}\left(\begin{array}{c}
2 k \\
k
\end{array}\right)\left(\begin{array}{c}
n+1 \\
k+1
\end{array}\right)\left(\begin{array}{c}
n+2 \\
k+1
\end{array}\right)
\]

证明Stanley-Wilf conjecture

The Stanley-Wilf conjecture

书里给出的思路是先丢个 The Füredi-Hajnal conjecture出来，说这个 The Füredi-Hajnal conjecture可以推导Stanley-Wilf conjecture.

这样我们先来研究The Füredi-Hajnal conjecture

The Füredi-Hajnal conjecture

\[f(n, P) \leq c_{p} n
\]

先空着

资料来自网络

书用的是Combinatorics of permutations by Miklos Bona