D4

# Dynkin diagram
 
       2
       |
   1---3---4
 
# coordinates (and numbering of) the positive co-roots

1   [1, 0, 0, 0]
2   [0, 1, 0, 0]
3   [0, 0, 1, 0]
4   [0, 0, 0, 1]
5   [1, 0, 1, 0]
6   [0, 1, 1, 0]
7   [0, 0, 1, 1]
8   [1, 1, 1, 0]
9   [1, 0, 1, 1]
10   [0, 1, 1, 1]
11   [1, 1, 1, 1]
12   [1, 1, 2, 1]

# f-vector of the cell complex, first segregated by faces of the
# dominant chamber, and then cumulative (q marks co-dimension):

[0, 0, 0, 0, 0]   {1, 2, 3, 4}
[0, 0, 0, 0, 1]   {2, 3, 4}
[0, 0, 0, 0, 1]   {1, 3, 4}
[0, 0, 0, 5, 1]   {3, 4}
[0, 0, 0, 0, 1]   {1, 2, 4}
[0, 0, 0, 3, 1]   {2, 4}
[0, 0, 0, 3, 1]   {1, 4}
[0, 0, 11, 7, 1]   {4}
[0, 0, 0, 0, 1]   {1, 2, 3}
[0, 0, 0, 5, 1]   {2, 3}
[0, 0, 0, 5, 1]   {1, 3}
[0, 0, 16, 9, 1]   {3}
[0, 0, 0, 3, 1]   {1, 2}
[0, 0, 11, 7, 1]   {2}
[0, 0, 11, 7, 1]   {1}
[50, 100, 66, 16, 1]   {}
                                        2       3       4
                      50 + 100 q + 115 q  + 70 q  + 15 q