书中基础篇很大的篇幅讨论了\(S_n(q)\)的一些渐进性质
续篇里试图说明\(S_n(q)\) how nice
多项式递归Polynomial Recursions
P-recursive和c-recursive定义
更多的内容看我的这篇播客园博客
例子：卡特兰数序列是P-recursive（或者说D-finite）
证明1
\[\sum_{n \geq 0} \frac{1}{n+1}\left(\begin{array}{c} 2 n \\ n \end{array}\right) x^{n}=\frac{2}{1 \pm \sqrt{1-4 x}} \]可以看到是algebraic,自然也是D-finite
证明2
\[(n+1)C_n-(4n-2)C_n-1=0 \]
两个说明\(S_n(q)\)和\(S_{n,r}(q)\)nice的推断
本篇用到的一些定义和记号
\(P\) be the infinite partially ordered set of all finite permutations ordered by pattern containment. 偏序关系是说\(p\le q\) 如果\(q\) contains \(p\) as a pattern
\(C\) a class consisting of finite permutations
\(C\) is a closed class of permutations <=> if \(q\in C\)且\(p\le q\) then \(p \in C\)
rational algebraic D-finite
更多的内容看我的这篇播客园博客
The P-recursiveness of \(S_{n,r}(132)\)
书中后面很大的篇幅都是在证明这个定理或者做准备工作
定义
front entries
Entries of an n-permutation p on the left of the entry n
back entries
those on the right of n
black entries
front entries中比【back entries最大】要小的元素构成 black entries
grey entries
back entries中比【front entries最小】要大的元素构成grey entries
图示
为什么我们给排列的元素染色？性质
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any black entry is smaller than any front entry which is not black， while any gray entry is larger than any back entry which is not gray.
前元素中，任何黑元素都要比非黑元素小；后元素中，任何灰元素都要比非灰元素大
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any black and any gray entry is part of at least one 132-subsequence
定义a 132-subsequence spans over the entry \(n\)
it starts with a front entry and ends with a back entry, then it must start with a black one and end with a gray one.
定义two permutations in the same strong class
amazing啊
定义
fundamental subsequence of \(p\)
两个class相似
The classes C and C’ are called similar if their permutations have fundamental subsequences of the same type.
same type
same type 是说
In the symmetric group \(S_n\), two permutations \(g\) and \(h\) are called conjugates of each other if there exists an element so that \(ƒgƒ^{-1}=h\) holds.
先写到这
资料来自网络
书用的是Combinatorics of permutations by Miklos Bona