先按行排序，再按列排序，则按行仍顺序

hbghlyj

先把每行排序，再把每列排序，则此时每行仍然是顺序的。
证明：
只需证明有$n$行、两列的情况：
设$f,g$是$\{1,2,\cdots,m\}$的置换，使得第一列$a_{f(n)}$顺序、第二列$b_{g(n)}$顺序。
假设第$1,\dots,n$行仍然顺序而第$n+1$行乱了：

根据左边,$a_{g(i)}<b_{g(i)}≤b_{g(n+1)}$对任何$i=1,2,\cdots,n+1$成立,所以
$$\{g(1),g(2),\cdots,g(n+1)\}\subseteq\{i∈S:a_i<b_{g(n+1)}\}\tag1$$
根据右边,
\begin{align*}
a_{f(1)}<a_{f(2)}<\cdots<a_{f(n)}&<b_{g(n)}\\
&<b_{g(n+1)}≤a_{f(n+1)}<a_{f(n+2)}<\cdots<a_{f(m)}
\end{align*}
所以$$\{f(1),f(2),\cdots,f(n)\}=\{i∈S:a_i<b_{g(n+1)}\}\tag2$$
由(1)与(2)得
$$\{g(1),g(2),\cdots,g(n+1)\}\subseteq\{f(1),f(2),\cdots,f(n)\}$$
左边比右边多1个元素.矛盾.
见此帖

hbghlyj

MathJax似乎不支持表格的高级语法, 如@{decl}
故采用i.upmath.me图片

hbghlyj

hbghlyj 发表于 2022-10-21 10:07
MathJax似乎不支持表格的高级语法, 如@{decl}

MathJax v4支持array环境的高级语法

New TeX Array Preamble Options

To help support line breaking within cells of wide tables, MathJax v4 includes support for the preamble column declarations defined in the array TeX package. These include the traditional c, l, and r for alignment of the contents of the cell (centered, left, or right), but adds support for p{width}, m{width}, and b{width} for vertical alignment of a fixed-width column in which line-breaking will occur at the given width, as well as w{align}{width} and W{align}{width}. There is also new support for >{...} and <{...} for adding content that is put before or after every entry in a column, as well as @{...} for replacing the inter-column space with the given content, and !{...} for replacing inter-column rules. Support for | and the non-standard : are improved so multiple copies of | and : now produce multiple rules that are close together. Finally, non-standard P{...}, M{...}, and B{...} are defined that produce math-mode versions of their corresponding lower-case counterparts. The \newcolumntype macro for declaring new column specifications is also available.

Note that for p, m, b, w, W, P, M, and B columns, line-breaking will occur to the given column with if line-breaking is the active overflow setting. Otherwise, wide content will overflow the width, as in actual LaTeX.