Computing dimensions

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I have the following routine divided into 3 steps:

1)

p = dict()
P = Permutations(3)
for i in (1 .. 3):
    for u in (1 .. 3):
        p[i, u] = [1 if P[m][i-1] == u else 0
                   for m in (0 .. 5)]

2)

for i in (1 .. 3):
     for u in (1 .. 3):
         for j in (1 .. 3):
             for v in (1 .. 3):
                 if i != j and u != v:
                     p[i, j, u, v] = [1 if P[m][i-1] == u and P[m][j-1] == v else 0
                                      for m in (0 .. 5)]

3)

A1 = matrix([p[i, u]
             for i in (1 .. 3)
             for u in (1 .. 3)])
A2 = matrix([p[i, j, u, v]
             for i in (1 .. 3) for j in (1 .. 3) if i != j
             for u in (1 .. 3) for v in (1 .. 3) if u != v])
C1 = A1.stack(A2)
E1 = A1.right_kernel()
E2 = C1.right_kernel()
d1 = E1.dimension()
d2 = E2.dimension()

I'm interested in dimensions d1 and d2.

If I go to permutations of 4, I will need to adjust all for loops to (1 .. 4) and to add one more step similar to step 2 to construct one more matrix like

A3 =  matrix([p[i, j, k, u, v, w]
              for i in (1 .. 4) for j in (1 .. 4)
              for u in (1 .. 4) for v in (1 .. 4)
              for k in (1 .. 4) for w in (1 .. 4)
              if i != j and i != k and j != k
              and u != v and u != w and v != w])

and also

C2 = C1.stack(A3)
E3 = C2.right_kernel()
d3 = E3.dimension()

So... I was wondering if there is some "easy" way to put this code in some kind of "generic" program. I mean, I could repeat the whole idea for permutations of 4 or 5 and so on, but it is not very practical. Any suggestions would be great.