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如何证明这道有关卡特兰数的恒等式?

任意2n-1个数之和的算式,添加n个左括号和n个右括号,要求

左、右括号是匹配的
算式开头有且仅有一个左括号,算式末尾有且仅有一个右括号
每个加号至多有一个相邻的括号
求证:用全部办法添加括号,各括号之积的和,等于每相邻的n个数之积的和。现举例如下:

当n=3时,算式1+23+45+678+9101,有以下5种匹配的括号串:((())),(()()),(())(),()(()),()()()

用全部5种办法添加括号:

(1+(23+(45)+678)+9101)

(1+(23)+45+(678)+9101)

(1+(23)+45)+678+(9101)

(1)+23+(45+(678)+9101)

(1)+23+(45)+678+(9101)

就有

(1+23+45+678+9101) ×(23+45+678)× 45+

(1+23+45+678+9101)× 23× 678+

(1+23+45)× 23 ×9101+

1× (45+678+9101)× 678+

1 ×45× 9101=505680576



(1+23+45)(23+45+678)(45+678+9101)= 505680576

平面几何中等价于托勒密定理的众多推广之一:

对于平面内任意凸2n边形,连接所有顶点且互不相交的n条线段之积的总和大于等于所有使得其两侧顶点个数相等的n条对角线乘积

现举例如下:
当n=3时,对于平面上任意六个点,有
AF•BE•CD + AF•BC•DE + EF•AD•BC + AB•CF•DE + AB•CD•EF ≥ AD•BE•CF
(例子的乘积顺序跟前面的是一致的),取等当且仅当六点共圆(或共线),称为Fuhrmann定理(《近代的欧氏几何学》)
v2-bef89908bc3ced88a716a0d8a6f248fe_b.jpg
2020-1-1 16:42

复数形式:(a-b)(f-c)(e-d)+(a-f)(b-e)(c-d)+(a-f)(b-c)(d-e)+(e-f)(a-d)(b-c)+(a-b)(e-f)(c-d)=(d-a)(b-e)(c-f)
这是我在知乎一年前的提问

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回复 2# isee

卡特兰数只是说明了括号串有多少个,但对恒等式的证明没啥用。

针对1#最后那条“复数形式”的恒等式,我的办法如下。

首先为了直观起见,将恒等式
\[(a_1-a_3)(a_2-a_4)=(a_1-a_2)(a_3-a_4)+(a_1-a_4)(a_2-a_3)\]用下图表示:
TIM截图20190801161340.png
2019-8-1 16:16

于是,对于六边形的情况,不断利用以上性质,将相交的变成不相交的之和,图示如下:
TIM截图20190801161617.png
2019-8-1 16:16

这样,相应的恒等式就是
\begin{align*}
(a_1-a_4)(a_2-a_5)(a_3-a_6)={}&(a_1-a_4)(a_2-a_3)(a_5-a_6)\\
&+(a_1-a_2)(a_3-a_4)(a_5-a_6)\\
&+(a_1-a_2)(a_3-a_6)(a_4-a_5)\\
&+(a_1-a_6)(a_2-a_5)(a_3-a_4)\\
&+(a_1-a_6)(a_2-a_3)(a_4-a_5).
\end{align*}
一般情况大概也是一样的,总可以变成所有不相交的积之和,从而得出结论。
$\href{https://kuingggg.github.io/}{\text{About Me}}$

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真的实操过才知道,定论还是下得太早了!
边数增加后,其实是有点不同的,来看看八边形的:
TIM截图20190801170646.png
2019-8-1 17:07

TIM截图20190801171801.png
2019-8-1 17:19

可以看到,最后会有两对是相同的,于是合并后就会有两个系数 2,故此恒等式为
  (a1 - a5)(a2 - a6)(a3 - a7)(a4 - a8)
=   (a1 - a4)(a2 - a3)(a5 - a8)(a6 - a7)
  + 2(a1 - a8)(a2 - a3)(a4 - a5)(a6 - a7)
  + (a1 - a2)(a3 - a8)(a4 - a7)(a5 - a6)
  + (a1 - a2)(a3 - a8)(a4 - a5)(a6 - a7)
  + 2(a1 - a2)(a3 - a4)(a5 - a6)(a7 - a8)
  + (a1 - a2)(a3 - a4)(a5 - a8)(a6 - a7)
  + (a1 - a4)(a2 - a3)(a5 - a6)(a7 - a8)
  + (a1 - a8)(a2 - a3)(a4 - a7)(a5 - a6)
  + (a1 - a2)(a3 - a6)(a4 - a5)(a7 - a8)
  + (a1 - a8)(a2 - a7)(a3 - a4)(a5 - a6)
  + (a1 - a8)(a2 - a7)(a3 - a6)(a4 - a5)
  + (a1 - a8)(a2 - a5)(a3 - a4)(a6 - a7)
  + (a1 - a6)(a2 - a3)(a4 - a5)(a7 - a8)
  + (a1 - a6)(a2 - a5)(a3 - a4)(a7 - a8)


相对应的几何命题就是:平面上任意八点 `A_1`, `A_2`,\ldots, `A_8`,有
`A_1A_5\cdot A_2A_6\cdot A_3A_7\cdot A_4A_8`
`{}\leqslant A_1A_4\cdot A_2A_3\cdot A_5A_8\cdot A_6A_7`
`{}+{\color{red}2}\cdot A_1A_8\cdot A_2A_3\cdot A_4A_5\cdot A_6A_7`
`{}+A_1A_2\cdot A_3A_8\cdot A_4A_7\cdot A_5A_6`
`{}+A_1A_2\cdot A_3A_8\cdot A_4A_5\cdot A_6A_7`
`{}+{\color{red}2}\cdot A_1A_2\cdot A_3A_4\cdot A_5A_6\cdot A_7A_8`
`{}+A_1A_2\cdot A_3A_4\cdot A_5A_8\cdot A_6A_7`
`{}+A_1A_4\cdot A_2A_3\cdot A_5A_6\cdot A_7A_8`
`{}+A_1A_8\cdot A_2A_3\cdot A_4A_7\cdot A_5A_6`
`{}+A_1A_2\cdot A_3A_6\cdot A_4A_5\cdot A_7A_8`
`{}+A_1A_8\cdot A_2A_7\cdot A_3A_4\cdot A_5A_6`
`{}+A_1A_8\cdot A_2A_7\cdot A_3A_6\cdot A_4A_5`
`{}+A_1A_8\cdot A_2A_5\cdot A_3A_4\cdot A_6A_7`
`{}+A_1A_6\cdot A_2A_3\cdot A_4A_5\cdot A_7A_8`
`{}+A_1A_6\cdot A_2A_5\cdot A_3A_4\cdot A_7A_8.`
$\href{https://kuingggg.github.io/}{\text{About Me}}$

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手工拆图实在太费神,于是接下的问题是,如何用软件自动生成这类恒等式?@编程大神

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回复 6# hbghlyj
n=4共线四点的欧拉定理就能写成轮换对称形式。所以这种也能写。用行列式的奇排列偶排列那一套能否解释?

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回复 6# hbghlyj

点那链接显示要登录知乎,我N年前好像注册过,但帐号啥的想不起来了,就没看了……

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回复 8# kuing
det($A_iA_j^2$)=0,其中$A_iA_i$=0,1≤i≤j≤n

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回复 11# hbghlyj

看不懂,或者能不能详细演示一下怎么生成 n=3 的 Fuhrmann定理?

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回复 9# hbghlyj

这个行列式有印象,应该很有名。找了一阵,后来在沈文选的《单形论导引》中翻到了,叫“平方距离矩阵”。四阶的时候是平面上四点共线或共圆的充要条件,五阶的时候是空间五点共面或共球的充要条件,……
但是这本书上面并没有说平面n点共圆时这个行列式的值为0.所以你的猜测大概率是错的。因为如若有这样漂亮的结论,相信应该很多人都应该知道了。。。

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平方距离矩阵
  1. With[{n = 2},Factor[Det[Table[(A @@ Sort[{i, j}])^2, {i, 2 n}, {j, 2 n}]
  2. /.A[i_, i_] -> 0]]]
复制代码
输出:
(A[1, 4] A[2, 3] - A[1, 3] A[2, 4] -A[1, 2] A[3, 4]) (A[1, 4] A[2, 3] + A[1, 3] A[2, 4] - A[1, 2] A[3, 4])
(A[1, 4] A[2, 3] - A[1, 3] A[2, 4] + A[1, 2] A[3, 4]) (A[1, 4] A[2, 3] + A[1, 3] A[2, 4] + A[1, 2] A[3, 4])

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回复 5# kuing
我不是编程大神,我只能半自动生成
n=3时
输入
  1. {{1, 4}, {2, 5}, {3,6}} //. {{t1___, {a_, b_}, t2___, {c_, d_}, t3___} /;
  2.     a < c < b < d -> {{t1, {a, c}, t2, {b, d}, t3}, {t1, {a, d},
  3.      t2, {c, b}, t3}}, {t1___, {a_, b_}, t2___, {c_, d_}, t3___} /;
  4.     c < a < d < b -> {{t1, {c, a}, t2, {d, b}, t3}, {t1, {a, d},
  5.      t2, {c, b}, t3}}}
复制代码
输出
{{{1, 2}, {4, 5}, {3,6}}, {{{{1, 2}, {3, 4}, {5, 6}}, {{1, 4}, {2, 3}, {5, 6}}}, {{{1,
     6}, {2, 3}, {4, 5}}, {{1, 6}, {2, 5}, {3, 4}}}}}
输入
  1. Partition[Partition[Flatten[%, 2], 3]
复制代码
输出
  1. {{{1, 2}, {4, 5}, {3, 6}}, {{1, 2}, {3, 4}, {5, 6}},
  2. {{1, 4}, {2,3}, {5, 6}}, {{1, 6}, {2, 3}, {4, 5}}, {{1, 6}, {2, 5}, {3, 4}}}
复制代码
输入
  1. Plus @@ Times @@@ (%58 /. {c_, d_} -> (Subscript[a, c] - Subscript[a, d]))
复制代码
输出$\left(a_3-a_4\right) \left(a_2-a_5\right) \left(a_1-a_6\right)+\left(a_2-a_3\right) \left(a_4-a_5\right) \left(a_1-a_6\right)+\left(a_1-a_2\right) \left(a_4-a_5\right) \left(a_3-a_6\right)+\left(a_2-a_3\right) \left(a_1-a_4\right) \left(a_5-a_6\right)+\left(a_1-a_2\right) \left(a_3-a_4\right) \left(a_5-a_6\right)$
验证:
输入Factor[%]
输出$-\left(a_4-a_1\right) \left(a_2-a_5\right) \left(a_3-a_6\right)$

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按照这个办法,n=4如下
  1. n = 4;
  2. table = Table[{i, i + n}, {i,
  3.      n}] //. {{t1___, {a_, b_}, t2___, {c_, d_}, t3___} /;
  4.       a < c < b < d -> {{t1, {a, c}, t2, {b, d}, t3}, {t1, {a, d},
  5.        t2, {c, b}, t3}}, {t1___, {a_, b_}, t2___, {c_, d_}, t3___} /;
  6.       c < a < d < b -> {{t1, {c, a}, t2, {d, b}, t3}, {t1, {a, d},
  7.        t2, {c, b}, t3}}};
  8. partition = Partition[Partition[Flatten[table], 2], n];
  9. expr = Plus @@
  10.   Times @@@ (partition /. {c_,
  11.        d_} -> (Subscript[a, c] - Subscript[a, d]))
  12. Factor[expr]
复制代码
输出$\left(a_4-a_5\right) \left(a_3-a_6\right) \left(a_2-a_7\right) \left(a_1-a_8\right)+\left(a_3-a_4\right) \left(a_5-a_6\right) \left(a_2-a_7\right) \left(a_1-a_8\right)+\left(a_2-a_3\right) \left(a_5-a_6\right) \left(a_4-a_7\right) \left(a_1-a_8\right)+\left(a_3-a_4\right) \left(a_2-a_5\right) \left(a_6-a_7\right) \left(a_1-a_8\right)+2 \left(a_2-a_3\right) \left(a_4-a_5\right) \left(a_6-a_7\right) \left(a_1-a_8\right)+\left(a_1-a_2\right) \left(a_5-a_6\right) \left(a_4-a_7\right) \left(a_3-a_8\right)+\left(a_1-a_2\right) \left(a_4-a_5\right) \left(a_6-a_7\right) \left(a_3-a_8\right)+\left(a_2-a_3\right) \left(a_1-a_4\right) \left(a_6-a_7\right) \left(a_5-a_8\right)+\left(a_1-a_2\right) \left(a_3-a_4\right) \left(a_6-a_7\right) \left(a_5-a_8\right)+\left(a_3-a_4\right) \left(a_2-a_5\right) \left(a_1-a_6\right) \left(a_7-a_8\right)+\left(a_2-a_3\right) \left(a_4-a_5\right) \left(a_1-a_6\right) \left(a_7-a_8\right)+\left(a_1-a_2\right) \left(a_4-a_5\right) \left(a_3-a_6\right) \left(a_7-a_8\right)+\left(a_2-a_3\right) \left(a_1-a_4\right) \left(a_5-a_6\right) \left(a_7-a_8\right)+2 \left(a_1-a_2\right) \left(a_3-a_4\right) \left(a_5-a_6\right) \left(a_7-a_8\right)$

$\left(a_5-a_1\right) \left(a_6-a_2\right) \left(a_3-a_7\right) \left(a_4-a_8\right)$

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n=5如下$\left(a_5-a_6\right) \left(a_4-a_7\right) \left(a_3-a_8\right) \left(a_2-a_9\right) \left(a_1-a_{10}\right)+\left(a_4-a_5\right) \left(a_6-a_7\right) \left(a_3-a_8\right) \left(a_2-a_9\right) \left(a_1-a_{10}\right)+\left(a_3-a_4\right) \left(a_6-a_7\right) \left(a_5-a_8\right) \left(a_2-a_9\right) \left(a_1-a_{10}\right)+\left(a_4-a_5\right) \left(a_3-a_6\right) \left(a_7-a_8\right) \left(a_2-a_9\right) \left(a_1-a_{10}\right)+2 \left(a_3-a_4\right) \left(a_5-a_6\right) \left(a_7-a_8\right) \left(a_2-a_9\right) \left(a_1-a_{10}\right)+\left(a_2-a_3\right) \left(a_6-a_7\right) \left(a_5-a_8\right) \left(a_4-a_9\right) \left(a_1-a_{10}\right)+2 \left(a_2-a_3\right) \left(a_5-a_6\right) \left(a_7-a_8\right) \left(a_4-a_9\right) \left(a_1-a_{10}\right)+\left(a_3-a_4\right) \left(a_2-a_5\right) \left(a_7-a_8\right) \left(a_6-a_9\right) \left(a_1-a_{10}\right)+2 \left(a_2-a_3\right) \left(a_4-a_5\right) \left(a_7-a_8\right) \left(a_6-a_9\right) \left(a_1-a_{10}\right)+\left(a_4-a_5\right) \left(a_3-a_6\right) \left(a_2-a_7\right) \left(a_8-a_9\right) \left(a_1-a_{10}\right)+2 \left(a_3-a_4\right) \left(a_5-a_6\right) \left(a_2-a_7\right) \left(a_8-a_9\right) \left(a_1-a_{10}\right)+2 \left(a_2-a_3\right) \left(a_5-a_6\right) \left(a_4-a_7\right) \left(a_8-a_9\right) \left(a_1-a_{10}\right)+2 \left(a_3-a_4\right) \left(a_2-a_5\right) \left(a_6-a_7\right) \left(a_8-a_9\right) \left(a_1-a_{10}\right)+3 \left(a_2-a_3\right) \left(a_4-a_5\right) \left(a_6-a_7\right) \left(a_8-a_9\right) \left(a_1-a_{10}\right)+\left(a_1-a_2\right) \left(a_6-a_7\right) \left(a_5-a_8\right) \left(a_4-a_9\right) \left(a_3-a_{10}\right)+\left(a_1-a_2\right) \left(a_5-a_6\right) \left(a_7-a_8\right) \left(a_4-a_9\right) \left(a_3-a_{10}\right)+\left(a_1-a_2\right) \left(a_4-a_5\right) \left(a_7-a_8\right) \left(a_6-a_9\right) \left(a_3-a_{10}\right)+\left(a_1-a_2\right) \left(a_5-a_6\right) \left(a_4-a_7\right) \left(a_8-a_9\right) \left(a_3-a_{10}\right)+2 \left(a_1-a_2\right) \left(a_4-a_5\right) \left(a_6-a_7\right) \left(a_8-a_9\right) \left(a_3-a_{10}\right)+\left(a_2-a_3\right) \left(a_1-a_4\right) \left(a_7-a_8\right) \left(a_6-a_9\right) \left(a_5-a_{10}\right)+\left(a_1-a_2\right) \left(a_3-a_4\right) \left(a_7-a_8\right) \left(a_6-a_9\right) \left(a_5-a_{10}\right)+\left(a_2-a_3\right) \left(a_1-a_4\right) \left(a_6-a_7\right) \left(a_8-a_9\right) \left(a_5-a_{10}\right)+2 \left(a_1-a_2\right) \left(a_3-a_4\right) \left(a_6-a_7\right) \left(a_8-a_9\right) \left(a_5-a_{10}\right)+\left(a_3-a_4\right) \left(a_2-a_5\right) \left(a_1-a_6\right) \left(a_8-a_9\right) \left(a_7-a_{10}\right)+\left(a_2-a_3\right) \left(a_4-a_5\right) \left(a_1-a_6\right) \left(a_8-a_9\right) \left(a_7-a_{10}\right)+\left(a_1-a_2\right) \left(a_4-a_5\right) \left(a_3-a_6\right) \left(a_8-a_9\right) \left(a_7-a_{10}\right)+\left(a_2-a_3\right) \left(a_1-a_4\right) \left(a_5-a_6\right) \left(a_8-a_9\right) \left(a_7-a_{10}\right)+2 \left(a_1-a_2\right) \left(a_3-a_4\right) \left(a_5-a_6\right) \left(a_8-a_9\right) \left(a_7-a_{10}\right)+\left(a_4-a_5\right) \left(a_3-a_6\right) \left(a_2-a_7\right) \left(a_1-a_8\right) \left(a_9-a_{10}\right)+\left(a_3-a_4\right) \left(a_5-a_6\right) \left(a_2-a_7\right) \left(a_1-a_8\right) \left(a_9-a_{10}\right)+\left(a_2-a_3\right) \left(a_5-a_6\right) \left(a_4-a_7\right) \left(a_1-a_8\right) \left(a_9-a_{10}\right)+\left(a_3-a_4\right) \left(a_2-a_5\right) \left(a_6-a_7\right) \left(a_1-a_8\right) \left(a_9-a_{10}\right)+2 \left(a_2-a_3\right) \left(a_4-a_5\right) \left(a_6-a_7\right) \left(a_1-a_8\right) \left(a_9-a_{10}\right)+\left(a_1-a_2\right) \left(a_5-a_6\right) \left(a_4-a_7\right) \left(a_3-a_8\right) \left(a_9-a_{10}\right)+2 \left(a_1-a_2\right) \left(a_4-a_5\right) \left(a_6-a_7\right) \left(a_3-a_8\right) \left(a_9-a_{10}\right)+\left(a_2-a_3\right) \left(a_1-a_4\right) \left(a_6-a_7\right) \left(a_5-a_8\right) \left(a_9-a_{10}\right)+2 \left(a_1-a_2\right) \left(a_3-a_4\right) \left(a_6-a_7\right) \left(a_5-a_8\right) \left(a_9-a_{10}\right)+\left(a_3-a_4\right) \left(a_2-a_5\right) \left(a_1-a_6\right) \left(a_7-a_8\right) \left(a_9-a_{10}\right)+2 \left(a_2-a_3\right) \left(a_4-a_5\right) \left(a_1-a_6\right) \left(a_7-a_8\right) \left(a_9-a_{10}\right)+2 \left(a_1-a_2\right) \left(a_4-a_5\right) \left(a_3-a_6\right) \left(a_7-a_8\right) \left(a_9-a_{10}\right)+2 \left(a_2-a_3\right) \left(a_1-a_4\right) \left(a_5-a_6\right) \left(a_7-a_8\right) \left(a_9-a_{10}\right)+3 \left(a_1-a_2\right) \left(a_3-a_4\right) \left(a_5-a_6\right) \left(a_7-a_8\right) \left(a_9-a_{10}\right)$

$\left(a_6-a_1\right) \left(a_7-a_2\right) \left(a_3-a_8\right) \left(a_4-a_9\right) \left(a_5-a_{10}\right)$

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n=6如下$\left(a_6-a_7\right) \left(a_5-a_8\right) \left(a_4-a_9\right) \left(a_3-a_{10}\right) \left(a_2-a_{11}\right) \left(a_1-a_{12}\right)+\left(a_5-a_6\right) \left(a_7-a_8\right) \left(a_4-a_9\right) \left(a_3-a_{10}\right) \left(a_2-a_{11}\right) \left(a_1-a_{12}\right)+\left(a_4-a_5\right) \left(a_7-a_8\right) \left(a_6-a_9\right) \left(a_3-a_{10}\right) \left(a_2-a_{11}\right) \left(a_1-a_{12}\right)+\left(a_5-a_6\right) \left(a_4-a_7\right) \left(a_8-a_9\right) \left(a_3-a_{10}\right) \left(a_2-a_{11}\right) \left(a_1-a_{12}\right)+2 \left(a_4-a_5\right) \left(a_6-a_7\right) \left(a_8-a_9\right) \left(a_3-a_{10}\right) \left(a_2-a_{11}\right) \left(a_1-a_{12}\right)+\left(a_3-a_4\right) \left(a_7-a_8\right) \left(a_6-a_9\right) \left(a_5-a_{10}\right) \left(a_2-a_{11}\right) \left(a_1-a_{12}\right)+2 \left(a_3-a_4\right) \left(a_6-a_7\right) \left(a_8-a_9\right) \left(a_5-a_{10}\right) \left(a_2-a_{11}\right) \left(a_1-a_{12}\right)+\left(a_4-a_5\right) \left(a_3-a_6\right) \left(a_8-a_9\right) \left(a_7-a_{10}\right) \left(a_2-a_{11}\right) \left(a_1-a_{12}\right)+2 \left(a_3-a_4\right) \left(a_5-a_6\right) \left(a_8-a_9\right) \left(a_7-a_{10}\right) \left(a_2-a_{11}\right) \left(a_1-a_{12}\right)+\left(a_5-a_6\right) \left(a_4-a_7\right) \left(a_3-a_8\right) \left(a_9-a_{10}\right) \left(a_2-a_{11}\right) \left(a_1-a_{12}\right)+2 \left(a_4-a_5\right) \left(a_6-a_7\right) \left(a_3-a_8\right) \left(a_9-a_{10}\right) \left(a_2-a_{11}\right) \left(a_1-a_{12}\right)+3 \left(a_3-a_4\right) \left(a_6-a_7\right) \left(a_5-a_8\right) \left(a_9-a_{10}\right) \left(a_2-a_{11}\right) \left(a_1-a_{12}\right)+2 \left(a_4-a_5\right) \left(a_3-a_6\right) \left(a_7-a_8\right) \left(a_9-a_{10}\right) \left(a_2-a_{11}\right) \left(a_1-a_{12}\right)+4 \left(a_3-a_4\right) \left(a_5-a_6\right) \left(a_7-a_8\right) \left(a_9-a_{10}\right) \left(a_2-a_{11}\right) \left(a_1-a_{12}\right)+\left(a_2-a_3\right) \left(a_7-a_8\right) \left(a_6-a_9\right) \left(a_5-a_{10}\right) \left(a_4-a_{11}\right) \left(a_1-a_{12}\right)+2 \left(a_2-a_3\right) \left(a_6-a_7\right) \left(a_8-a_9\right) \left(a_5-a_{10}\right) \left(a_4-a_{11}\right) \left(a_1-a_{12}\right)+2 \left(a_2-a_3\right) \left(a_5-a_6\right) \left(a_8-a_9\right) \left(a_7-a_{10}\right) \left(a_4-a_{11}\right) \left(a_1-a_{12}\right)+2 \left(a_2-a_3\right) \left(a_6-a_7\right) \left(a_5-a_8\right) \left(a_9-a_{10}\right) \left(a_4-a_{11}\right) \left(a_1-a_{12}\right)+3 \left(a_2-a_3\right) \left(a_5-a_6\right) \left(a_7-a_8\right) \left(a_9-a_{10}\right) \left(a_4-a_{11}\right) \left(a_1-a_{12}\right)+\left(a_3-a_4\right) \left(a_2-a_5\right) \left(a_8-a_9\right) \left(a_7-a_{10}\right) \left(a_6-a_{11}\right) \left(a_1-a_{12}\right)+2 \left(a_2-a_3\right) \left(a_4-a_5\right) \left(a_8-a_9\right) \left(a_7-a_{10}\right) \left(a_6-a_{11}\right) \left(a_1-a_{12}\right)+2 \left(a_3-a_4\right) \left(a_2-a_5\right) \left(a_7-a_8\right) \left(a_9-a_{10}\right) \left(a_6-a_{11}\right) \left(a_1-a_{12}\right)+3 \left(a_2-a_3\right) \left(a_4-a_5\right) \left(a_7-a_8\right) \left(a_9-a_{10}\right) \left(a_6-a_{11}\right) \left(a_1-a_{12}\right)+\left(a_4-a_5\right) \left(a_3-a_6\right) \left(a_2-a_7\right) \left(a_9-a_{10}\right) \left(a_8-a_{11}\right) \left(a_1-a_{12}\right)+2 \left(a_3-a_4\right) \left(a_5-a_6\right) \left(a_2-a_7\right) \left(a_9-a_{10}\right) \left(a_8-a_{11}\right) \left(a_1-a_{12}\right)+2 \left(a_2-a_3\right) \left(a_5-a_6\right) \left(a_4-a_7\right) \left(a_9-a_{10}\right) \left(a_8-a_{11}\right) \left(a_1-a_{12}\right)+3 \left(a_3-a_4\right) \left(a_2-a_5\right) \left(a_6-a_7\right) \left(a_9-a_{10}\right) \left(a_8-a_{11}\right) \left(a_1-a_{12}\right)+4 \left(a_2-a_3\right) \left(a_4-a_5\right) \left(a_6-a_7\right) \left(a_9-a_{10}\right) \left(a_8-a_{11}\right) \left(a_1-a_{12}\right)+\left(a_5-a_6\right) \left(a_4-a_7\right) \left(a_3-a_8\right) \left(a_2-a_9\right) \left(a_{10}-a_{11}\right) \left(a_1-a_{12}\right)+2 \left(a_4-a_5\right) \left(a_6-a_7\right) \left(a_3-a_8\right) \left(a_2-a_9\right) \left(a_{10}-a_{11}\right) \left(a_1-a_{12}\right)+2 \left(a_3-a_4\right) \left(a_6-a_7\right) \left(a_5-a_8\right) \left(a_2-a_9\right) \left(a_{10}-a_{11}\right) \left(a_1-a_{12}\right)+2 \left(a_4-a_5\right) \left(a_3-a_6\right) \left(a_7-a_8\right) \left(a_2-a_9\right) \left(a_{10}-a_{11}\right) \left(a_1-a_{12}\right)+3 \left(a_3-a_4\right) \left(a_5-a_6\right) \left(a_7-a_8\right) \left(a_2-a_9\right) \left(a_{10}-a_{11}\right) \left(a_1-a_{12}\right)+2 \left(a_2-a_3\right) \left(a_6-a_7\right) \left(a_5-a_8\right) \left(a_4-a_9\right) \left(a_{10}-a_{11}\right) \left(a_1-a_{12}\right)+3 \left(a_2-a_3\right) \left(a_5-a_6\right) \left(a_7-a_8\right) \left(a_4-a_9\right) \left(a_{10}-a_{11}\right) \left(a_1-a_{12}\right)+2 \left(a_3-a_4\right) \left(a_2-a_5\right) \left(a_7-a_8\right) \left(a_6-a_9\right) \left(a_{10}-a_{11}\right) \left(a_1-a_{12}\right)+4 \left(a_2-a_3\right) \left(a_4-a_5\right) \left(a_7-a_8\right) \left(a_6-a_9\right) \left(a_{10}-a_{11}\right) \left(a_1-a_{12}\right)+2 \left(a_4-a_5\right) \left(a_3-a_6\right) \left(a_2-a_7\right) \left(a_8-a_9\right) \left(a_{10}-a_{11}\right) \left(a_1-a_{12}\right)+3 \left(a_3-a_4\right) \left(a_5-a_6\right) \left(a_2-a_7\right) \left(a_8-a_9\right) \left(a_{10}-a_{11}\right) \left(a_1-a_{12}\right)+4 \left(a_2-a_3\right) \left(a_5-a_6\right) \left(a_4-a_7\right) \left(a_8-a_9\right) \left(a_{10}-a_{11}\right) \left(a_1-a_{12}\right)+4 \left(a_3-a_4\right) \left(a_2-a_5\right) \left(a_6-a_7\right) \left(a_8-a_9\right) \left(a_{10}-a_{11}\right) \left(a_1-a_{12}\right)+8 \left(a_2-a_3\right) \left(a_4-a_5\right) \left(a_6-a_7\right) \left(a_8-a_9\right) \left(a_{10}-a_{11}\right) \left(a_1-a_{12}\right)+\left(a_1-a_2\right) \left(a_7-a_8\right) \left(a_6-a_9\right) \left(a_5-a_{10}\right) \left(a_4-a_{11}\right) \left(a_3-a_{12}\right)+\left(a_1-a_2\right) \left(a_6-a_7\right) \left(a_8-a_9\right) \left(a_5-a_{10}\right) \left(a_4-a_{11}\right) \left(a_3-a_{12}\right)+\left(a_1-a_2\right) \left(a_5-a_6\right) \left(a_8-a_9\right) \left(a_7-a_{10}\right) \left(a_4-a_{11}\right) \left(a_3-a_{12}\right)+\left(a_1-a_2\right) \left(a_6-a_7\right) \left(a_5-a_8\right) \left(a_9-a_{10}\right) \left(a_4-a_{11}\right) \left(a_3-a_{12}\right)+2 \left(a_1-a_2\right) \left(a_5-a_6\right) \left(a_7-a_8\right) \left(a_9-a_{10}\right) \left(a_4-a_{11}\right) \left(a_3-a_{12}\right)+\left(a_1-a_2\right) \left(a_4-a_5\right) \left(a_8-a_9\right) \left(a_7-a_{10}\right) \left(a_6-a_{11}\right) \left(a_3-a_{12}\right)+2 \left(a_1-a_2\right) \left(a_4-a_5\right) \left(a_7-a_8\right) \left(a_9-a_{10}\right) \left(a_6-a_{11}\right) \left(a_3-a_{12}\right)+\left(a_1-a_2\right) \left(a_5-a_6\right) \left(a_4-a_7\right) \left(a_9-a_{10}\right) \left(a_8-a_{11}\right) \left(a_3-a_{12}\right)+2 \left(a_1-a_2\right) \left(a_4-a_5\right) \left(a_6-a_7\right) \left(a_9-a_{10}\right) \left(a_8-a_{11}\right) \left(a_3-a_{12}\right)+\left(a_1-a_2\right) \left(a_6-a_7\right) \left(a_5-a_8\right) \left(a_4-a_9\right) \left(a_{10}-a_{11}\right) \left(a_3-a_{12}\right)+2 \left(a_1-a_2\right) \left(a_5-a_6\right) \left(a_7-a_8\right) \left(a_4-a_9\right) \left(a_{10}-a_{11}\right) \left(a_3-a_{12}\right)+3 \left(a_1-a_2\right) \left(a_4-a_5\right) \left(a_7-a_8\right) \left(a_6-a_9\right) \left(a_{10}-a_{11}\right) \left(a_3-a_{12}\right)+2 \left(a_1-a_2\right) \left(a_5-a_6\right) \left(a_4-a_7\right) \left(a_8-a_9\right) \left(a_{10}-a_{11}\right) \left(a_3-a_{12}\right)+4 \left(a_1-a_2\right) \left(a_4-a_5\right) \left(a_6-a_7\right) \left(a_8-a_9\right) \left(a_{10}-a_{11}\right) \left(a_3-a_{12}\right)+\left(a_2-a_3\right) \left(a_1-a_4\right) \left(a_8-a_9\right) \left(a_7-a_{10}\right) \left(a_6-a_{11}\right) \left(a_5-a_{12}\right)+\left(a_1-a_2\right) \left(a_3-a_4\right) \left(a_8-a_9\right) \left(a_7-a_{10}\right) \left(a_6-a_{11}\right) \left(a_5-a_{12}\right)+\left(a_2-a_3\right) \left(a_1-a_4\right) \left(a_7-a_8\right) \left(a_9-a_{10}\right) \left(a_6-a_{11}\right) \left(a_5-a_{12}\right)+2 \left(a_1-a_2\right) \left(a_3-a_4\right) \left(a_7-a_8\right) \left(a_9-a_{10}\right) \left(a_6-a_{11}\right) \left(a_5-a_{12}\right)+\left(a_2-a_3\right) \left(a_1-a_4\right) \left(a_6-a_7\right) \left(a_9-a_{10}\right) \left(a_8-a_{11}\right) \left(a_5-a_{12}\right)+2 \left(a_1-a_2\right) \left(a_3-a_4\right) \left(a_6-a_7\right) \left(a_9-a_{10}\right) \left(a_8-a_{11}\right) \left(a_5-a_{12}\right)+\left(a_2-a_3\right) \left(a_1-a_4\right) \left(a_7-a_8\right) \left(a_6-a_9\right) \left(a_{10}-a_{11}\right) \left(a_5-a_{12}\right)+2 \left(a_1-a_2\right) \left(a_3-a_4\right) \left(a_7-a_8\right) \left(a_6-a_9\right) \left(a_{10}-a_{11}\right) \left(a_5-a_{12}\right)+2 \left(a_2-a_3\right) \left(a_1-a_4\right) \left(a_6-a_7\right) \left(a_8-a_9\right) \left(a_{10}-a_{11}\right) \left(a_5-a_{12}\right)+3 \left(a_1-a_2\right) \left(a_3-a_4\right) \left(a_6-a_7\right) \left(a_8-a_9\right) \left(a_{10}-a_{11}\right) \left(a_5-a_{12}\right)+\left(a_3-a_4\right) \left(a_2-a_5\right) \left(a_1-a_6\right) \left(a_9-a_{10}\right) \left(a_8-a_{11}\right) \left(a_7-a_{12}\right)+\left(a_2-a_3\right) \left(a_4-a_5\right) \left(a_1-a_6\right) \left(a_9-a_{10}\right) \left(a_8-a_{11}\right) \left(a_7-a_{12}\right)+\left(a_1-a_2\right) \left(a_4-a_5\right) \left(a_3-a_6\right) \left(a_9-a_{10}\right) \left(a_8-a_{11}\right) \left(a_7-a_{12}\right)+\left(a_2-a_3\right) \left(a_1-a_4\right) \left(a_5-a_6\right) \left(a_9-a_{10}\right) \left(a_8-a_{11}\right) \left(a_7-a_{12}\right)+2 \left(a_1-a_2\right) \left(a_3-a_4\right) \left(a_5-a_6\right) \left(a_9-a_{10}\right) \left(a_8-a_{11}\right) \left(a_7-a_{12}\right)+\left(a_3-a_4\right) \left(a_2-a_5\right) \left(a_1-a_6\right) \left(a_8-a_9\right) \left(a_{10}-a_{11}\right) \left(a_7-a_{12}\right)+2 \left(a_2-a_3\right) \left(a_4-a_5\right) \left(a_1-a_6\right) \left(a_8-a_9\right) \left(a_{10}-a_{11}\right) \left(a_7-a_{12}\right)+2 \left(a_1-a_2\right) \left(a_4-a_5\right) \left(a_3-a_6\right) \left(a_8-a_9\right) \left(a_{10}-a_{11}\right) \left(a_7-a_{12}\right)+2 \left(a_2-a_3\right) \left(a_1-a_4\right) \left(a_5-a_6\right) \left(a_8-a_9\right) \left(a_{10}-a_{11}\right) \left(a_7-a_{12}\right)+3 \left(a_1-a_2\right) \left(a_3-a_4\right) \left(a_5-a_6\right) \left(a_8-a_9\right) \left(a_{10}-a_{11}\right) \left(a_7-a_{12}\right)+\left(a_4-a_5\right) \left(a_3-a_6\right) \left(a_2-a_7\right) \left(a_1-a_8\right) \left(a_{10}-a_{11}\right) \left(a_9-a_{12}\right)+\left(a_3-a_4\right) \left(a_5-a_6\right) \left(a_2-a_7\right) \left(a_1-a_8\right) \left(a_{10}-a_{11}\right) \left(a_9-a_{12}\right)+\left(a_2-a_3\right) \left(a_5-a_6\right) \left(a_4-a_7\right) \left(a_1-a_8\right) \left(a_{10}-a_{11}\right) \left(a_9-a_{12}\right)+\left(a_3-a_4\right) \left(a_2-a_5\right) \left(a_6-a_7\right) \left(a_1-a_8\right) \left(a_{10}-a_{11}\right) \left(a_9-a_{12}\right)+2 \left(a_2-a_3\right) \left(a_4-a_5\right) \left(a_6-a_7\right) \left(a_1-a_8\right) \left(a_{10}-a_{11}\right) \left(a_9-a_{12}\right)+\left(a_1-a_2\right) \left(a_5-a_6\right) \left(a_4-a_7\right) \left(a_3-a_8\right) \left(a_{10}-a_{11}\right) \left(a_9-a_{12}\right)+2 \left(a_1-a_2\right) \left(a_4-a_5\right) \left(a_6-a_7\right) \left(a_3-a_8\right) \left(a_{10}-a_{11}\right) \left(a_9-a_{12}\right)+\left(a_2-a_3\right) \left(a_1-a_4\right) \left(a_6-a_7\right) \left(a_5-a_8\right) \left(a_{10}-a_{11}\right) \left(a_9-a_{12}\right)+2 \left(a_1-a_2\right) \left(a_3-a_4\right) \left(a_6-a_7\right) \left(a_5-a_8\right) \left(a_{10}-a_{11}\right) \left(a_9-a_{12}\right)+\left(a_3-a_4\right) \left(a_2-a_5\right) \left(a_1-a_6\right) \left(a_7-a_8\right) \left(a_{10}-a_{11}\right) \left(a_9-a_{12}\right)+2 \left(a_2-a_3\right) \left(a_4-a_5\right) \left(a_1-a_6\right) \left(a_7-a_8\right) \left(a_{10}-a_{11}\right) \left(a_9-a_{12}\right)+3 \left(a_1-a_2\right) \left(a_4-a_5\right) \left(a_3-a_6\right) \left(a_7-a_8\right) \left(a_{10}-a_{11}\right) \left(a_9-a_{12}\right)+2 \left(a_2-a_3\right) \left(a_1-a_4\right) \left(a_5-a_6\right) \left(a_7-a_8\right) \left(a_{10}-a_{11}\right) \left(a_9-a_{12}\right)+4 \left(a_1-a_2\right) \left(a_3-a_4\right) \left(a_5-a_6\right) \left(a_7-a_8\right) \left(a_{10}-a_{11}\right) \left(a_9-a_{12}\right)+\left(a_5-a_6\right) \left(a_4-a_7\right) \left(a_3-a_8\right) \left(a_2-a_9\right) \left(a_1-a_{10}\right) \left(a_{11}-a_{12}\right)+\left(a_4-a_5\right) \left(a_6-a_7\right) \left(a_3-a_8\right) \left(a_2-a_9\right) \left(a_1-a_{10}\right) \left(a_{11}-a_{12}\right)+\left(a_3-a_4\right) \left(a_6-a_7\right) \left(a_5-a_8\right) \left(a_2-a_9\right) \left(a_1-a_{10}\right) \left(a_{11}-a_{12}\right)+\left(a_4-a_5\right) \left(a_3-a_6\right) \left(a_7-a_8\right) \left(a_2-a_9\right) \left(a_1-a_{10}\right) \left(a_{11}-a_{12}\right)+2 \left(a_3-a_4\right) \left(a_5-a_6\right) \left(a_7-a_8\right) \left(a_2-a_9\right) \left(a_1-a_{10}\right) \left(a_{11}-a_{12}\right)+\left(a_2-a_3\right) \left(a_6-a_7\right) \left(a_5-a_8\right) \left(a_4-a_9\right) \left(a_1-a_{10}\right) \left(a_{11}-a_{12}\right)+2 \left(a_2-a_3\right) \left(a_5-a_6\right) \left(a_7-a_8\right) \left(a_4-a_9\right) \left(a_1-a_{10}\right) \left(a_{11}-a_{12}\right)+\left(a_3-a_4\right) \left(a_2-a_5\right) \left(a_7-a_8\right) \left(a_6-a_9\right) \left(a_1-a_{10}\right) \left(a_{11}-a_{12}\right)+2 \left(a_2-a_3\right) \left(a_4-a_5\right) \left(a_7-a_8\right) \left(a_6-a_9\right) \left(a_1-a_{10}\right) \left(a_{11}-a_{12}\right)+\left(a_4-a_5\right) \left(a_3-a_6\right) \left(a_2-a_7\right) \left(a_8-a_9\right) \left(a_1-a_{10}\right) \left(a_{11}-a_{12}\right)+2 \left(a_3-a_4\right) \left(a_5-a_6\right) \left(a_2-a_7\right) \left(a_8-a_9\right) \left(a_1-a_{10}\right) \left(a_{11}-a_{12}\right)+3 \left(a_2-a_3\right) \left(a_5-a_6\right) \left(a_4-a_7\right) \left(a_8-a_9\right) \left(a_1-a_{10}\right) \left(a_{11}-a_{12}\right)+2 \left(a_3-a_4\right) \left(a_2-a_5\right) \left(a_6-a_7\right) \left(a_8-a_9\right) \left(a_1-a_{10}\right) \left(a_{11}-a_{12}\right)+4 \left(a_2-a_3\right) \left(a_4-a_5\right) \left(a_6-a_7\right) \left(a_8-a_9\right) \left(a_1-a_{10}\right) \left(a_{11}-a_{12}\right)+\left(a_1-a_2\right) \left(a_6-a_7\right) \left(a_5-a_8\right) \left(a_4-a_9\right) \left(a_3-a_{10}\right) \left(a_{11}-a_{12}\right)+2 \left(a_1-a_2\right) \left(a_5-a_6\right) \left(a_7-a_8\right) \left(a_4-a_9\right) \left(a_3-a_{10}\right) \left(a_{11}-a_{12}\right)+2 \left(a_1-a_2\right) \left(a_4-a_5\right) \left(a_7-a_8\right) \left(a_6-a_9\right) \left(a_3-a_{10}\right) \left(a_{11}-a_{12}\right)+2 \left(a_1-a_2\right) \left(a_5-a_6\right) \left(a_4-a_7\right) \left(a_8-a_9\right) \left(a_3-a_{10}\right) \left(a_{11}-a_{12}\right)+3 \left(a_1-a_2\right) \left(a_4-a_5\right) \left(a_6-a_7\right) \left(a_8-a_9\right) \left(a_3-a_{10}\right) \left(a_{11}-a_{12}\right)+\left(a_2-a_3\right) \left(a_1-a_4\right) \left(a_7-a_8\right) \left(a_6-a_9\right) \left(a_5-a_{10}\right) \left(a_{11}-a_{12}\right)+2 \left(a_1-a_2\right) \left(a_3-a_4\right) \left(a_7-a_8\right) \left(a_6-a_9\right) \left(a_5-a_{10}\right) \left(a_{11}-a_{12}\right)+2 \left(a_2-a_3\right) \left(a_1-a_4\right) \left(a_6-a_7\right) \left(a_8-a_9\right) \left(a_5-a_{10}\right) \left(a_{11}-a_{12}\right)+3 \left(a_1-a_2\right) \left(a_3-a_4\right) \left(a_6-a_7\right) \left(a_8-a_9\right) \left(a_5-a_{10}\right) \left(a_{11}-a_{12}\right)+\left(a_3-a_4\right) \left(a_2-a_5\right) \left(a_1-a_6\right) \left(a_8-a_9\right) \left(a_7-a_{10}\right) \left(a_{11}-a_{12}\right)+2 \left(a_2-a_3\right) \left(a_4-a_5\right) \left(a_1-a_6\right) \left(a_8-a_9\right) \left(a_7-a_{10}\right) \left(a_{11}-a_{12}\right)+2 \left(a_1-a_2\right) \left(a_4-a_5\right) \left(a_3-a_6\right) \left(a_8-a_9\right) \left(a_7-a_{10}\right) \left(a_{11}-a_{12}\right)+3 \left(a_2-a_3\right) \left(a_1-a_4\right) \left(a_5-a_6\right) \left(a_8-a_9\right) \left(a_7-a_{10}\right) \left(a_{11}-a_{12}\right)+4 \left(a_1-a_2\right) \left(a_3-a_4\right) \left(a_5-a_6\right) \left(a_8-a_9\right) \left(a_7-a_{10}\right) \left(a_{11}-a_{12}\right)+\left(a_4-a_5\right) \left(a_3-a_6\right) \left(a_2-a_7\right) \left(a_1-a_8\right) \left(a_9-a_{10}\right) \left(a_{11}-a_{12}\right)+2 \left(a_3-a_4\right) \left(a_5-a_6\right) \left(a_2-a_7\right) \left(a_1-a_8\right) \left(a_9-a_{10}\right) \left(a_{11}-a_{12}\right)+2 \left(a_2-a_3\right) \left(a_5-a_6\right) \left(a_4-a_7\right) \left(a_1-a_8\right) \left(a_9-a_{10}\right) \left(a_{11}-a_{12}\right)+2 \left(a_3-a_4\right) \left(a_2-a_5\right) \left(a_6-a_7\right) \left(a_1-a_8\right) \left(a_9-a_{10}\right) \left(a_{11}-a_{12}\right)+3 \left(a_2-a_3\right) \left(a_4-a_5\right) \left(a_6-a_7\right) \left(a_1-a_8\right) \left(a_9-a_{10}\right) \left(a_{11}-a_{12}\right)+2 \left(a_1-a_2\right) \left(a_5-a_6\right) \left(a_4-a_7\right) \left(a_3-a_8\right) \left(a_9-a_{10}\right) \left(a_{11}-a_{12}\right)+3 \left(a_1-a_2\right) \left(a_4-a_5\right) \left(a_6-a_7\right) \left(a_3-a_8\right) \left(a_9-a_{10}\right) \left(a_{11}-a_{12}\right)+2 \left(a_2-a_3\right) \left(a_1-a_4\right) \left(a_6-a_7\right) \left(a_5-a_8\right) \left(a_9-a_{10}\right) \left(a_{11}-a_{12}\right)+4 \left(a_1-a_2\right) \left(a_3-a_4\right) \left(a_6-a_7\right) \left(a_5-a_8\right) \left(a_9-a_{10}\right) \left(a_{11}-a_{12}\right)+2 \left(a_3-a_4\right) \left(a_2-a_5\right) \left(a_1-a_6\right) \left(a_7-a_8\right) \left(a_9-a_{10}\right) \left(a_{11}-a_{12}\right)+3 \left(a_2-a_3\right) \left(a_4-a_5\right) \left(a_1-a_6\right) \left(a_7-a_8\right) \left(a_9-a_{10}\right) \left(a_{11}-a_{12}\right)+4 \left(a_1-a_2\right) \left(a_4-a_5\right) \left(a_3-a_6\right) \left(a_7-a_8\right) \left(a_9-a_{10}\right) \left(a_{11}-a_{12}\right)+4 \left(a_2-a_3\right) \left(a_1-a_4\right) \left(a_5-a_6\right) \left(a_7-a_8\right) \left(a_9-a_{10}\right) \left(a_{11}-a_{12}\right)+8 \left(a_1-a_2\right) \left(a_3-a_4\right) \left(a_5-a_6\right) \left(a_7-a_8\right) \left(a_9-a_{10}\right) \left(a_{11}-a_{12}\right)$

$-\left(a_7-a_1\right) \left(a_8-a_2\right) \left(a_9-a_3\right) \left(a_4-a_{10}\right) \left(a_5-a_{11}\right) \left(a_6-a_{12}\right)$

TOP

n=7如下$\left(a_7-a_8\right) \left(a_6-a_9\right) \left(a_5-a_{10}\right) \left(a_4-a_{11}\right) \left(a_3-a_{12}\right) \left(a_2-a_{13}\right) \left(a_1-a_{14}\right)+\left(a_6-a_7\right) \left(a_8-a_9\right) \left(a_5-a_{10}\right) \left(a_4-a_{11}\right) \left(a_3-a_{12}\right) \left(a_2-a_{13}\right) \left(a_1-a_{14}\right)+\left(a_5-a_6\right) \left(a_8-a_9\right) \left(a_7-a_{10}\right) \left(a_4-a_{11}\right) \left(a_3-a_{12}\right) \left(a_2-a_{13}\right) \left(a_1-a_{14}\right)+\left(a_6-a_7\right) \left(a_5-a_8\right) \left(a_9-a_{10}\right) \left(a_4-a_{11}\right) \left(a_3-a_{12}\right) \left(a_2-a_{13}\right) \left(a_1-a_{14}\right)+2 \left(a_5-a_6\right) \left(a_7-a_8\right) \left(a_9-a_{10}\right) \left(a_4-a_{11}\right) \left(a_3-a_{12}\right) \left(a_2-a_{13}\right) \left(a_1-a_{14}\right)+\left(a_4-a_5\right) \left(a_8-a_9\right) \left(a_7-a_{10}\right) \left(a_6-a_{11}\right) \left(a_3-a_{12}\right) \left(a_2-a_{13}\right) \left(a_1-a_{14}\right)+2 \left(a_4-a_5\right) \left(a_7-a_8\right) \left(a_9-a_{10}\right) \left(a_6-a_{11}\right) \left(a_3-a_{12}\right) \left(a_2-a_{13}\right) \left(a_1-a_{14}\right)+\left(a_5-a_6\right) \left(a_4-a_7\right) \left(a_9-a_{10}\right) \left(a_8-a_{11}\right) \left(a_3-a_{12}\right) \left(a_2-a_{13}\right) \left(a_1-a_{14}\right)+2 \left(a_4-a_5\right) \left(a_6-a_7\right) \left(a_9-a_{10}\right) \left(a_8-a_{11}\right) \left(a_3-a_{12}\right) \left(a_2-a_{13}\right) \left(a_1-a_{14}\right)+\left(a_6-a_7\right) \left(a_5-a_8\right) \left(a_4-a_9\right) \left(a_{10}-a_{11}\right) \left(a_3-a_{12}\right) \left(a_2-a_{13}\right) \left(a_1-a_{14}\right)+2 \left(a_5-a_6\right) \left(a_7-a_8\right) \left(a_4-a_9\right) \left(a_{10}-a_{11}\right) \left(a_3-a_{12}\right) \left(a_2-a_{13}\right) \left(a_1-a_{14}\right)+3 \left(a_4-a_5\right) \left(a_7-a_8\right) \left(a_6-a_9\right) \left(a_{10}-a_{11}\right) \left(a_3-a_{12}\right) \left(a_2-a_{13}\right) \left(a_1-a_{14}\right)+2 \left(a_5-a_6\right) \left(a_4-a_7\right) \left(a_8-a_9\right) \left(a_{10}-a_{11}\right) \left(a_3-a_{12}\right) \left(a_2-a_{13}\right) \left(a_1-a_{14}\right)+4 \left(a_4-a_5\right) \left(a_6-a_7\right) \left(a_8-a_9\right) \left(a_{10}-a_{11}\right) \left(a_3-a_{12}\right) \left(a_2-a_{13}\right) \left(a_1-a_{14}\right)+\left(a_3-a_4\right) \left(a_8-a_9\right) \left(a_7-a_{10}\right) \left(a_6-a_{11}\right) \left(a_5-a_{12}\right) \left(a_2-a_{13}\right) \left(a_1-a_{14}\right)+2 \left(a_3-a_4\right) \left(a_7-a_8\right) \left(a_9-a_{10}\right) \left(a_6-a_{11}\right) \left(a_5-a_{12}\right) \left(a_2-a_{13}\right) \left(a_1-a_{14}\right)+2 \left(a_3-a_4\right) \left(a_6-a_7\right) \left(a_9-a_{10}\right) \left(a_8-a_{11}\right) \left(a_5-a_{12}\right) \left(a_2-a_{13}\right) \left(a_1-a_{14}\right)+3 \left(a_3-a_4\right) \left(a_7-a_8\right) \left(a_6-a_9\right) \left(a_{10}-a_{11}\right) \left(a_5-a_{12}\right) \left(a_2-a_{13}\right) \left(a_1-a_{14}\right)+4 \left(a_3-a_4\right) \left(a_6-a_7\right) \left(a_8-a_9\right) \left(a_{10}-a_{11}\right) \left(a_5-a_{12}\right) \left(a_2-a_{13}\right) \left(a_1-a_{14}\right)+\left(a_4-a_5\right) \left(a_3-a_6\right) \left(a_9-a_{10}\right) \left(a_8-a_{11}\right) \left(a_7-a_{12}\right) \left(a_2-a_{13}\right) \left(a_1-a_{14}\right)+2 \left(a_3-a_4\right) \left(a_5-a_6\right) \left(a_9-a_{10}\right) \left(a_8-a_{11}\right) \left(a_7-a_{12}\right) \left(a_2-a_{13}\right) \left(a_1-a_{14}\right)+2 \left(a_4-a_5\right) \left(a_3-a_6\right) \left(a_8-a_9\right) \left(a_{10}-a_{11}\right) \left(a_7-a_{12}\right) \left(a_2-a_{13}\right) \left(a_1-a_{14}\right)+4 \left(a_3-a_4\right) \left(a_5-a_6\right) \left(a_8-a_9\right) \left(a_{10}-a_{11}\right) \left(a_7-a_{12}\right) \left(a_2-a_{13}\right) \left(a_1-a_{14}\right)+\left(a_5-a_6\right) \left(a_4-a_7\right) \left(a_3-a_8\right) \left(a_{10}-a_{11}\right) \left(a_9-a_{12}\right) \left(a_2-a_{13}\right) \left(a_1-a_{14}\right)+2 \left(a_4-a_5\right) \left(a_6-a_7\right) \left(a_3-a_8\right) \left(a_{10}-a_{11}\right) \left(a_9-a_{12}\right) \left(a_2-a_{13}\right) \left(a_1-a_{14}\right)+3 \left(a_3-a_4\right) \left(a_6-a_7\right) \left(a_5-a_8\right) \left(a_{10}-a_{11}\right) \left(a_9-a_{12}\right) \left(a_2-a_{13}\right) \left(a_1-a_{14}\right)+3 \left(a_4-a_5\right) \left(a_3-a_6\right) \left(a_7-a_8\right) \left(a_{10}-a_{11}\right) \left(a_9-a_{12}\right) \left(a_2-a_{13}\right) \left(a_1-a_{14}\right)+6 \left(a_3-a_4\right) \left(a_5-a_6\right) \left(a_7-a_8\right) \left(a_{10}-a_{11}\right) \left(a_9-a_{12}\right) \left(a_2-a_{13}\right) \left(a_1-a_{14}\right)+\left(a_6-a_7\right) \left(a_5-a_8\right) \left(a_4-a_9\right) \left(a_3-a_{10}\right) \left(a_{11}-a_{12}\right) \left(a_2-a_{13}\right) \left(a_1-a_{14}\right)+2 \left(a_5-a_6\right) \left(a_7-a_8\right) \left(a_4-a_9\right) \left(a_3-a_{10}\right) \left(a_{11}-a_{12}\right) \left(a_2-a_{13}\right) \left(a_1-a_{14}\right)+3 \left(a_4-a_5\right) \left(a_7-a_8\right) \left(a_6-a_9\right) \left(a_3-a_{10}\right) \left(a_{11}-a_{12}\right) \left(a_2-a_{13}\right) \left(a_1-a_{14}\right)+2 \left(a_5-a_6\right) \left(a_4-a_7\right) \left(a_8-a_9\right) \left(a_3-a_{10}\right) \left(a_{11}-a_{12}\right) \left(a_2-a_{13}\right) \left(a_1-a_{14}\right)+4 \left(a_4-a_5\right) \left(a_6-a_7\right) \left(a_8-a_9\right) \left(a_3-a_{10}\right) \left(a_{11}-a_{12}\right) \left(a_2-a_{13}\right) \left(a_1-a_{14}\right)+3 \left(a_3-a_4\right) \left(a_7-a_8\right) \left(a_6-a_9\right) \left(a_5-a_{10}\right) \left(a_{11}-a_{12}\right) \left(a_2-a_{13}\right) \left(a_1-a_{14}\right)+4 \left(a_3-a_4\right) \left(a_6-a_7\right) \left(a_8-a_9\right) \left(a_5-a_{10}\right) \left(a_{11}-a_{12}\right) \left(a_2-a_{13}\right) \left(a_1-a_{14}\right)+3 \left(a_4-a_5\right) \left(a_3-a_6\right) \left(a_8-a_9\right) \left(a_7-a_{10}\right) \left(a_{11}-a_{12}\right) \left(a_2-a_{13}\right) \left(a_1-a_{14}\right)+5 \left(a_3-a_4\right) \left(a_5-a_6\right) \left(a_8-a_9\right) \left(a_7-a_{10}\right) \left(a_{11}-a_{12}\right) \left(a_2-a_{13}\right) \left(a_1-a_{14}\right)+2 \left(a_5-a_6\right) \left(a_4-a_7\right) \left(a_3-a_8\right) \left(a_9-a_{10}\right) \left(a_{11}-a_{12}\right) \left(a_2-a_{13}\right) \left(a_1-a_{14}\right)+4 \left(a_4-a_5\right) \left(a_6-a_7\right) \left(a_3-a_8\right) \left(a_9-a_{10}\right) \left(a_{11}-a_{12}\right) \left(a_2-a_{13}\right) \left(a_1-a_{14}\right)+5 \left(a_3-a_4\right) \left(a_6-a_7\right) \left(a_5-a_8\right) \left(a_9-a_{10}\right) \left(a_{11}-a_{12}\right) \left(a_2-a_{13}\right) \left(a_1-a_{14}\right)+6 \left(a_4-a_5\right) \left(a_3-a_6\right) \left(a_7-a_8\right) \left(a_9-a_{10}\right) \left(a_{11}-a_{12}\right) \left(a_2-a_{13}\right) \left(a_1-a_{14}\right)+10 \left(a_3-a_4\right) \left(a_5-a_6\right) \left(a_7-a_8\right) \left(a_9-a_{10}\right) \left(a_{11}-a_{12}\right) \left(a_2-a_{13}\right) \left(a_1-a_{14}\right)+\left(a_2-a_3\right) \left(a_8-a_9\right) \left(a_7-a_{10}\right) \left(a_6-a_{11}\right) \left(a_5-a_{12}\right) \left(a_4-a_{13}\right) \left(a_1-a_{14}\right)+2 \left(a_2-a_3\right) \left(a_7-a_8\right) \left(a_9-a_{10}\right) \left(a_6-a_{11}\right) \left(a_5-a_{12}\right) \left(a_4-a_{13}\right) \left(a_1-a_{14}\right)+2 \left(a_2-a_3\right) \left(a_6-a_7\right) \left(a_9-a_{10}\right) \left(a_8-a_{11}\right) \left(a_5-a_{12}\right) \left(a_4-a_{13}\right) \left(a_1-a_{14}\right)+2 \left(a_2-a_3\right) \left(a_7-a_8\right) \left(a_6-a_9\right) \left(a_{10}-a_{11}\right) \left(a_5-a_{12}\right) \left(a_4-a_{13}\right) \left(a_1-a_{14}\right)+3 \left(a_2-a_3\right) \left(a_6-a_7\right) \left(a_8-a_9\right) \left(a_{10}-a_{11}\right) \left(a_5-a_{12}\right) \left(a_4-a_{13}\right) \left(a_1-a_{14}\right)+2 \left(a_2-a_3\right) \left(a_5-a_6\right) \left(a_9-a_{10}\right) \left(a_8-a_{11}\right) \left(a_7-a_{12}\right) \left(a_4-a_{13}\right) \left(a_1-a_{14}\right)+3 \left(a_2-a_3\right) \left(a_5-a_6\right) 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\left(a_{13}-a_{14}\right)+3 \left(a_3-a_4\right) \left(a_5-a_6\right) \left(a_2-a_7\right) \left(a_8-a_9\right) \left(a_1-a_{10}\right) \left(a_{11}-a_{12}\right) \left(a_{13}-a_{14}\right)+4 \left(a_2-a_3\right) \left(a_5-a_6\right) \left(a_4-a_7\right) \left(a_8-a_9\right) \left(a_1-a_{10}\right) \left(a_{11}-a_{12}\right) \left(a_{13}-a_{14}\right)+4 \left(a_3-a_4\right) \left(a_2-a_5\right) \left(a_6-a_7\right) \left(a_8-a_9\right) \left(a_1-a_{10}\right) \left(a_{11}-a_{12}\right) \left(a_{13}-a_{14}\right)+8 \left(a_2-a_3\right) \left(a_4-a_5\right) \left(a_6-a_7\right) \left(a_8-a_9\right) \left(a_1-a_{10}\right) \left(a_{11}-a_{12}\right) \left(a_{13}-a_{14}\right)+2 \left(a_1-a_2\right) \left(a_6-a_7\right) \left(a_5-a_8\right) \left(a_4-a_9\right) \left(a_3-a_{10}\right) \left(a_{11}-a_{12}\right) \left(a_{13}-a_{14}\right)+3 \left(a_1-a_2\right) \left(a_5-a_6\right) \left(a_7-a_8\right) \left(a_4-a_9\right) \left(a_3-a_{10}\right) \left(a_{11}-a_{12}\right) \left(a_{13}-a_{14}\right)+4 \left(a_1-a_2\right) \left(a_4-a_5\right) \left(a_7-a_8\right) \left(a_6-a_9\right) \left(a_3-a_{10}\right) \left(a_{11}-a_{12}\right) \left(a_{13}-a_{14}\right)+4 \left(a_1-a_2\right) \left(a_5-a_6\right) \left(a_4-a_7\right) \left(a_8-a_9\right) \left(a_3-a_{10}\right) \left(a_{11}-a_{12}\right) \left(a_{13}-a_{14}\right)+8 \left(a_1-a_2\right) \left(a_4-a_5\right) \left(a_6-a_7\right) \left(a_8-a_9\right) \left(a_3-a_{10}\right) \left(a_{11}-a_{12}\right) \left(a_{13}-a_{14}\right)+2 \left(a_2-a_3\right) \left(a_1-a_4\right) \left(a_7-a_8\right) \left(a_6-a_9\right) \left(a_5-a_{10}\right) \left(a_{11}-a_{12}\right) \left(a_{13}-a_{14}\right)+4 \left(a_1-a_2\right) \left(a_3-a_4\right) \left(a_7-a_8\right) \left(a_6-a_9\right) \left(a_5-a_{10}\right) \left(a_{11}-a_{12}\right) \left(a_{13}-a_{14}\right)+4 \left(a_2-a_3\right) \left(a_1-a_4\right) \left(a_6-a_7\right) \left(a_8-a_9\right) \left(a_5-a_{10}\right) \left(a_{11}-a_{12}\right) \left(a_{13}-a_{14}\right)+8 \left(a_1-a_2\right) \left(a_3-a_4\right) \left(a_6-a_7\right) \left(a_8-a_9\right) \left(a_5-a_{10}\right) \left(a_{11}-a_{12}\right) \left(a_{13}-a_{14}\right)+2 \left(a_3-a_4\right) \left(a_2-a_5\right) \left(a_1-a_6\right) \left(a_8-a_9\right) \left(a_7-a_{10}\right) \left(a_{11}-a_{12}\right) \left(a_{13}-a_{14}\right)+4 \left(a_2-a_3\right) \left(a_4-a_5\right) \left(a_1-a_6\right) \left(a_8-a_9\right) \left(a_7-a_{10}\right) \left(a_{11}-a_{12}\right) \left(a_{13}-a_{14}\right)+6 \left(a_1-a_2\right) \left(a_4-a_5\right) \left(a_3-a_6\right) \left(a_8-a_9\right) \left(a_7-a_{10}\right) \left(a_{11}-a_{12}\right) \left(a_{13}-a_{14}\right)+5 \left(a_2-a_3\right) \left(a_1-a_4\right) \left(a_5-a_6\right) \left(a_8-a_9\right) \left(a_7-a_{10}\right) \left(a_{11}-a_{12}\right) \left(a_{13}-a_{14}\right)+10 \left(a_1-a_2\right) \left(a_3-a_4\right) \left(a_5-a_6\right) \left(a_8-a_9\right) \left(a_7-a_{10}\right) \left(a_{11}-a_{12}\right) \left(a_{13}-a_{14}\right)+2 \left(a_4-a_5\right) \left(a_3-a_6\right) \left(a_2-a_7\right) \left(a_1-a_8\right) \left(a_9-a_{10}\right) \left(a_{11}-a_{12}\right) \left(a_{13}-a_{14}\right)+3 \left(a_3-a_4\right) \left(a_5-a_6\right) \left(a_2-a_7\right) \left(a_1-a_8\right) \left(a_9-a_{10}\right) \left(a_{11}-a_{12}\right) \left(a_{13}-a_{14}\right)+4 \left(a_2-a_3\right) \left(a_5-a_6\right) \left(a_4-a_7\right) \left(a_1-a_8\right) \left(a_9-a_{10}\right) \left(a_{11}-a_{12}\right) \left(a_{13}-a_{14}\right)+4 \left(a_3-a_4\right) \left(a_2-a_5\right) \left(a_6-a_7\right) \left(a_1-a_8\right) \left(a_9-a_{10}\right) \left(a_{11}-a_{12}\right) \left(a_{13}-a_{14}\right)+8 \left(a_2-a_3\right) \left(a_4-a_5\right) \left(a_6-a_7\right) \left(a_1-a_8\right) \left(a_9-a_{10}\right) \left(a_{11}-a_{12}\right) \left(a_{13}-a_{14}\right)+4 \left(a_1-a_2\right) \left(a_5-a_6\right) \left(a_4-a_7\right) \left(a_3-a_8\right) \left(a_9-a_{10}\right) \left(a_{11}-a_{12}\right) \left(a_{13}-a_{14}\right)+8 \left(a_1-a_2\right) \left(a_4-a_5\right) \left(a_6-a_7\right) \left(a_3-a_8\right) \left(a_9-a_{10}\right) \left(a_{11}-a_{12}\right) \left(a_{13}-a_{14}\right)+6 \left(a_2-a_3\right) \left(a_1-a_4\right) \left(a_6-a_7\right) \left(a_5-a_8\right) \left(a_9-a_{10}\right) \left(a_{11}-a_{12}\right) \left(a_{13}-a_{14}\right)+10 \left(a_1-a_2\right) \left(a_3-a_4\right) \left(a_6-a_7\right) \left(a_5-a_8\right) \left(a_9-a_{10}\right) \left(a_{11}-a_{12}\right) \left(a_{13}-a_{14}\right)+4 \left(a_3-a_4\right) \left(a_2-a_5\right) \left(a_1-a_6\right) \left(a_7-a_8\right) \left(a_9-a_{10}\right) \left(a_{11}-a_{12}\right) \left(a_{13}-a_{14}\right)+8 \left(a_2-a_3\right) \left(a_4-a_5\right) \left(a_1-a_6\right) \left(a_7-a_8\right) \left(a_9-a_{10}\right) \left(a_{11}-a_{12}\right) \left(a_{13}-a_{14}\right)+10 \left(a_1-a_2\right) \left(a_4-a_5\right) \left(a_3-a_6\right) \left(a_7-a_8\right) \left(a_9-a_{10}\right) \left(a_{11}-a_{12}\right) \left(a_{13}-a_{14}\right)+10 \left(a_2-a_3\right) \left(a_1-a_4\right) \left(a_5-a_6\right) \left(a_7-a_8\right) \left(a_9-a_{10}\right) \left(a_{11}-a_{12}\right) \left(a_{13}-a_{14}\right)+17 \left(a_1-a_2\right) \left(a_3-a_4\right) \left(a_5-a_6\right) \left(a_7-a_8\right) \left(a_9-a_{10}\right) \left(a_{11}-a_{12}\right) \left(a_{13}-a_{14}\right)$

$-\left(a_8-a_1\right) \left(a_9-a_2\right) \left(a_{10}-a_3\right) \left(a_4-a_{11}\right) \left(a_5-a_{12}\right) \left(a_6-a_{13}\right) \left(a_7-a_{14}\right)$

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到此为止吧.我也找不出什么规律来

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回复 17# hbghlyj

已经很牛比了,浏览器都要卡爆了

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找到了一个规律:
$(a_1-a_2)(a_3-a_4)\cdots(a_{2n-1}-a_{2n})$的系数是最大的.
\begin{matrix}
n&2&3&4&5&6&7\\
最大系数&1&1&2&3&8&17
\end{matrix}
https://oeis.org/search?q=1%2C1% ... glish&go=Search
有待进一步研究

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