单形之积的剖分

hbghlyj

$p,q\inN_+,$
$\Delta^p$是$p$维单形
$\Delta^q$是$q$维单形
则$\Delta^p\times\Delta^q$是很多$\Delta^{p+q}$粘合得到的。具体有多少

hbghlyj

$\Delta^1$是$0\to 1$
那麼$\Delta^1\times \Delta^1$可分为2个$\Delta^2$:
$$(0,0)\to(1,0)\to(1,1)$$
$$(0,0)\to(0,1)\to(1,1)$$

hbghlyj

$\Delta^1$是$0\to 1$
$\Delta^2$是$0\to 1\to2$
那麼$\Delta^1\times \Delta^2$可分为3个$\Delta^3$:
$$(0,0)\to(0,1)\to(0,2)\to(1,2)$$
$$(0,0)\to(0,1)\to(1,1)\to(1,2)$$
$$(0,0)\to(1,0)\to(1,1)\to(1,2)$$

hbghlyj

可以用体积计算：
$\Delta^p$的$p$维体积是$\frac1{p!}$
$\Delta^q$的$q$维体积是$\frac1{q!}$
那么$\Delta^p\times\Delta^q$的($p+q$)维体积是$\frac1{p!}\cdot\frac1{q!}$
而$\Delta^{p+q}$的($p+q$)维体积是$\frac1{(p+q)!}$
那么$\Delta^p\times\Delta^q$分为$\frac{(p+q)!}{p!q!}=\binom{p+q}p$个$\Delta^{p+q}$
个数与(p,q)-shuffle相同

个数确定了。但怎么写出它们的頂点呢

hbghlyj

hbghlyj

hbghlyj

hbghlyj 发表于 2024-8-25 13:02
例如$3\times 2$网格上的一条阶梯路径可以被描绘成

这幅图来自 Gabriel, Zisman, "Calculus of Fractions and Homotopy Theory", chapter II

hbghlyj

$\Delta^p\times\Delta^q\times\Delta^r$分为多少个$\Delta^{p+q+r}$
按“体积”计算应该是$\dfrac{(p+q+r)!}{p!q!r!}$
但是我们可以像上面一样用顶点的形式写出来吗？

hbghlyj

1. def find_paths(x, y, path, paths):
2.     if x == 3 and y == 2:
3.         paths.append(path)
4.         return
5.     if x < 3:
6.         find_paths(x + 1, y, path + [(x + 1, y)], paths)
7.     if y < 2:
8.         find_paths(x, y + 1, path + [(x, y + 1)], paths)
10. paths = []
11. find_paths(0, 0, [(0, 0)], paths)
12. for path in paths:
13.     print("\\to".join(map(str, path)))

复制代码

\begin{aligned}(0, 0)\to(1, 0)\to(2, 0)\to(3, 0)\to(3, 1)\to(3, 2)\\
(0, 0)\to(1, 0)\to(2, 0)\to(2, 1)\to(3, 1)\to(3, 2)\\
(0, 0)\to(1, 0)\to(2, 0)\to(2, 1)\to(2, 2)\to(3, 2)\\
(0, 0)\to(1, 0)\to(1, 1)\to(2, 1)\to(3, 1)\to(3, 2)\\
(0, 0)\to(1, 0)\to(1, 1)\to(2, 1)\to(2, 2)\to(3, 2)\\
(0, 0)\to(1, 0)\to(1, 1)\to(1, 2)\to(2, 2)\to(3, 2)\\
(0, 0)\to(0, 1)\to(1, 1)\to(2, 1)\to(3, 1)\to(3, 2)\\
(0, 0)\to(0, 1)\to(1, 1)\to(2, 1)\to(2, 2)\to(3, 2)\\
(0, 0)\to(0, 1)\to(1, 1)\to(1, 2)\to(2, 2)\to(3, 2)\\
(0, 0)\to(0, 1)\to(0, 2)\to(1, 2)\to(2, 2)\to(3, 2)\end{aligned}

hbghlyj

1. from itertools import permutations
3. def generate_paths():
4.     steps = ['x', 'y', 'z', 'w', 'w']
5.    
6.     unique_permutations = set(permutations(steps))
7.    
8.     paths = []
9.     for perm in unique_permutations:
10.         path = [(0, 0, 0, 0)]
11.         current_position = [0, 0, 0, 0]
12.         for step in perm:
13.             if step == 'x':
14.                 current_position[0] += 1
15.             elif step == 'y':
16.                 current_position[1] += 1
17.             elif step == 'z':
18.                 current_position[2] += 1
19.             elif step == 'w':
20.                 current_position[3] += 1
21.             path.append(tuple(current_position))
22.         paths.append(path)
23.    
24.     return paths
26. paths = generate_paths()
27. for path in paths:
28.     print("\\to".join(map(str, path)))

复制代码

\begin{align*}
(0, 0, 0, 0)\to(1, 0, 0, 0)\to(1, 0, 1, 0)\to(1, 0, 1, 1)\to(1, 1, 1, 1)\to(1, 1, 1, 2)\\
(0, 0, 0, 0)\to(0, 1, 0, 0)\to(0, 1, 0, 1)\to(1, 1, 0, 1)\to(1, 1, 0, 2)\to(1, 1, 1, 2)\\
(0, 0, 0, 0)\to(0, 0, 0, 1)\to(1, 0, 0, 1)\to(1, 1, 0, 1)\to(1, 1, 0, 2)\to(1, 1, 1, 2)\\
(0, 0, 0, 0)\to(1, 0, 0, 0)\to(1, 1, 0, 0)\to(1, 1, 1, 0)\to(1, 1, 1, 1)\to(1, 1, 1, 2)\\
(0, 0, 0, 0)\to(0, 1, 0, 0)\to(0, 1, 1, 0)\to(0, 1, 1, 1)\to(1, 1, 1, 1)\to(1, 1, 1, 2)\\
(0, 0, 0, 0)\to(0, 1, 0, 0)\to(1, 1, 0, 0)\to(1, 1, 1, 0)\to(1, 1, 1, 1)\to(1, 1, 1, 2)\\
(0, 0, 0, 0)\to(1, 0, 0, 0)\to(1, 0, 1, 0)\to(1, 0, 1, 1)\to(1, 0, 1, 2)\to(1, 1, 1, 2)\\
(0, 0, 0, 0)\to(0, 0, 0, 1)\to(0, 0, 1, 1)\to(0, 1, 1, 1)\to(0, 1, 1, 2)\to(1, 1, 1, 2)\\
(0, 0, 0, 0)\to(0, 0, 1, 0)\to(0, 0, 1, 1)\to(1, 0, 1, 1)\to(1, 1, 1, 1)\to(1, 1, 1, 2)\\
(0, 0, 0, 0)\to(0, 0, 0, 1)\to(0, 0, 1, 1)\to(1, 0, 1, 1)\to(1, 0, 1, 2)\to(1, 1, 1, 2)\\
(0, 0, 0, 0)\to(1, 0, 0, 0)\to(1, 0, 0, 1)\to(1, 0, 0, 2)\to(1, 0, 1, 2)\to(1, 1, 1, 2)\\
(0, 0, 0, 0)\to(1, 0, 0, 0)\to(1, 0, 0, 1)\to(1, 1, 0, 1)\to(1, 1, 1, 1)\to(1, 1, 1, 2)\\
(0, 0, 0, 0)\to(0, 0, 1, 0)\to(0, 1, 1, 0)\to(1, 1, 1, 0)\to(1, 1, 1, 1)\to(1, 1, 1, 2)\\
(0, 0, 0, 0)\to(0, 0, 0, 1)\to(0, 1, 0, 1)\to(0, 1, 1, 1)\to(0, 1, 1, 2)\to(1, 1, 1, 2)\\
(0, 0, 0, 0)\to(0, 0, 0, 1)\to(0, 0, 0, 2)\to(1, 0, 0, 2)\to(1, 0, 1, 2)\to(1, 1, 1, 2)\\
(0, 0, 0, 0)\to(0, 1, 0, 0)\to(0, 1, 0, 1)\to(0, 1, 1, 1)\to(0, 1, 1, 2)\to(1, 1, 1, 2)\\
(0, 0, 0, 0)\to(0, 0, 0, 1)\to(0, 0, 0, 2)\to(0, 1, 0, 2)\to(1, 1, 0, 2)\to(1, 1, 1, 2)\\
(0, 0, 0, 0)\to(0, 0, 1, 0)\to(0, 0, 1, 1)\to(1, 0, 1, 1)\to(1, 0, 1, 2)\to(1, 1, 1, 2)\\
(0, 0, 0, 0)\to(0, 0, 0, 1)\to(0, 1, 0, 1)\to(1, 1, 0, 1)\to(1, 1, 1, 1)\to(1, 1, 1, 2)\\
(0, 0, 0, 0)\to(0, 0, 0, 1)\to(0, 1, 0, 1)\to(0, 1, 1, 1)\to(1, 1, 1, 1)\to(1, 1, 1, 2)\\
(0, 0, 0, 0)\to(0, 0, 1, 0)\to(1, 0, 1, 0)\to(1, 0, 1, 1)\to(1, 1, 1, 1)\to(1, 1, 1, 2)\\
(0, 0, 0, 0)\to(0, 0, 0, 1)\to(0, 0, 1, 1)\to(0, 0, 1, 2)\to(0, 1, 1, 2)\to(1, 1, 1, 2)\\
(0, 0, 0, 0)\to(0, 1, 0, 0)\to(0, 1, 0, 1)\to(0, 1, 1, 1)\to(1, 1, 1, 1)\to(1, 1, 1, 2)\\
(0, 0, 0, 0)\to(0, 0, 0, 1)\to(1, 0, 0, 1)\to(1, 0, 0, 2)\to(1, 0, 1, 2)\to(1, 1, 1, 2)\\
(0, 0, 0, 0)\to(1, 0, 0, 0)\to(1, 0, 0, 1)\to(1, 1, 0, 1)\to(1, 1, 0, 2)\to(1, 1, 1, 2)\\
(0, 0, 0, 0)\to(0, 0, 0, 1)\to(0, 0, 0, 2)\to(0, 0, 1, 2)\to(1, 0, 1, 2)\to(1, 1, 1, 2)\\
(0, 0, 0, 0)\to(1, 0, 0, 0)\to(1, 0, 0, 1)\to(1, 0, 0, 2)\to(1, 1, 0, 2)\to(1, 1, 1, 2)\\
(0, 0, 0, 0)\to(0, 0, 1, 0)\to(0, 0, 1, 1)\to(0, 0, 1, 2)\to(0, 1, 1, 2)\to(1, 1, 1, 2)\\
(0, 0, 0, 0)\to(0, 0, 0, 1)\to(0, 0, 0, 2)\to(1, 0, 0, 2)\to(1, 1, 0, 2)\to(1, 1, 1, 2)\\
(0, 0, 0, 0)\to(0, 0, 0, 1)\to(0, 1, 0, 1)\to(0, 1, 0, 2)\to(0, 1, 1, 2)\to(1, 1, 1, 2)\\
(0, 0, 0, 0)\to(0, 0, 0, 1)\to(1, 0, 0, 1)\to(1, 0, 0, 2)\to(1, 1, 0, 2)\to(1, 1, 1, 2)\\
(0, 0, 0, 0)\to(0, 0, 0, 1)\to(0, 0, 1, 1)\to(0, 1, 1, 1)\to(1, 1, 1, 1)\to(1, 1, 1, 2)\\
(0, 0, 0, 0)\to(1, 0, 0, 0)\to(1, 0, 0, 1)\to(1, 0, 1, 1)\to(1, 0, 1, 2)\to(1, 1, 1, 2)\\
(0, 0, 0, 0)\to(0, 1, 0, 0)\to(0, 1, 0, 1)\to(0, 1, 0, 2)\to(0, 1, 1, 2)\to(1, 1, 1, 2)\\
(0, 0, 0, 0)\to(0, 0, 1, 0)\to(1, 0, 1, 0)\to(1, 0, 1, 1)\to(1, 0, 1, 2)\to(1, 1, 1, 2)\\
(0, 0, 0, 0)\to(1, 0, 0, 0)\to(1, 0, 0, 1)\to(1, 0, 1, 1)\to(1, 1, 1, 1)\to(1, 1, 1, 2)\\
(0, 0, 0, 0)\to(0, 0, 1, 0)\to(0, 1, 1, 0)\to(0, 1, 1, 1)\to(0, 1, 1, 2)\to(1, 1, 1, 2)\\
(0, 0, 0, 0)\to(0, 0, 0, 1)\to(0, 1, 0, 1)\to(1, 1, 0, 1)\to(1, 1, 0, 2)\to(1, 1, 1, 2)\\
(0, 0, 0, 0)\to(0, 0, 1, 0)\to(0, 0, 1, 1)\to(0, 0, 1, 2)\to(1, 0, 1, 2)\to(1, 1, 1, 2)\\
(0, 0, 0, 0)\to(0, 1, 0, 0)\to(1, 1, 0, 0)\to(1, 1, 0, 1)\to(1, 1, 1, 1)\to(1, 1, 1, 2)\\
(0, 0, 0, 0)\to(0, 0, 1, 0)\to(0, 0, 1, 1)\to(0, 1, 1, 1)\to(0, 1, 1, 2)\to(1, 1, 1, 2)\\
(0, 0, 0, 0)\to(0, 0, 0, 1)\to(1, 0, 0, 1)\to(1, 0, 1, 1)\to(1, 1, 1, 1)\to(1, 1, 1, 2)\\
(0, 0, 0, 0)\to(0, 0, 0, 1)\to(0, 0, 1, 1)\to(0, 0, 1, 2)\to(1, 0, 1, 2)\to(1, 1, 1, 2)\\
(0, 0, 0, 0)\to(0, 0, 1, 0)\to(0, 1, 1, 0)\to(0, 1, 1, 1)\to(1, 1, 1, 1)\to(1, 1, 1, 2)\\
(0, 0, 0, 0)\to(0, 0, 0, 1)\to(1, 0, 0, 1)\to(1, 0, 1, 1)\to(1, 0, 1, 2)\to(1, 1, 1, 2)\\
(0, 0, 0, 0)\to(0, 0, 1, 0)\to(0, 0, 1, 1)\to(0, 1, 1, 1)\to(1, 1, 1, 1)\to(1, 1, 1, 2)\\
(0, 0, 0, 0)\to(1, 0, 0, 0)\to(1, 0, 1, 0)\to(1, 1, 1, 0)\to(1, 1, 1, 1)\to(1, 1, 1, 2)\\
(0, 0, 0, 0)\to(0, 1, 0, 0)\to(0, 1, 0, 1)\to(1, 1, 0, 1)\to(1, 1, 1, 1)\to(1, 1, 1, 2)\\
(0, 0, 0, 0)\to(0, 1, 0, 0)\to(0, 1, 1, 0)\to(1, 1, 1, 0)\to(1, 1, 1, 1)\to(1, 1, 1, 2)\\
(0, 0, 0, 0)\to(0, 1, 0, 0)\to(1, 1, 0, 0)\to(1, 1, 0, 1)\to(1, 1, 0, 2)\to(1, 1, 1, 2)\\
(0, 0, 0, 0)\to(1, 0, 0, 0)\to(1, 1, 0, 0)\to(1, 1, 0, 1)\to(1, 1, 1, 1)\to(1, 1, 1, 2)\\
(0, 0, 0, 0)\to(0, 0, 1, 0)\to(1, 0, 1, 0)\to(1, 1, 1, 0)\to(1, 1, 1, 1)\to(1, 1, 1, 2)\\
(0, 0, 0, 0)\to(0, 0, 0, 1)\to(0, 0, 0, 2)\to(0, 1, 0, 2)\to(0, 1, 1, 2)\to(1, 1, 1, 2)\\
(0, 0, 0, 0)\to(0, 1, 0, 0)\to(0, 1, 1, 0)\to(0, 1, 1, 1)\to(0, 1, 1, 2)\to(1, 1, 1, 2)\\
(0, 0, 0, 0)\to(0, 0, 0, 1)\to(0, 1, 0, 1)\to(0, 1, 0, 2)\to(1, 1, 0, 2)\to(1, 1, 1, 2)\\
(0, 0, 0, 0)\to(0, 0, 0, 1)\to(1, 0, 0, 1)\to(1, 1, 0, 1)\to(1, 1, 1, 1)\to(1, 1, 1, 2)\\
(0, 0, 0, 0)\to(0, 0, 0, 1)\to(0, 0, 1, 1)\to(1, 0, 1, 1)\to(1, 1, 1, 1)\to(1, 1, 1, 2)\\
(0, 0, 0, 0)\to(0, 1, 0, 0)\to(0, 1, 0, 1)\to(0, 1, 0, 2)\to(1, 1, 0, 2)\to(1, 1, 1, 2)\\
(0, 0, 0, 0)\to(1, 0, 0, 0)\to(1, 1, 0, 0)\to(1, 1, 0, 1)\to(1, 1, 0, 2)\to(1, 1, 1, 2)\\
(0, 0, 0, 0)\to(0, 0, 0, 1)\to(0, 0, 0, 2)\to(0, 0, 1, 2)\to(0, 1, 1, 2)\to(1, 1, 1, 2)\end{align*}