Can this be done with Polya-Theoy?

  Given the graph of the dodecahedron.
  (20 vertices, 30 edges, 3-regular).

  - How many different orientations of 
    the edges are possible for a given
    in-degree vector (i0, i1, i2, i3)?


(Where i_k is the number of vertices with
in-degree k.)

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The Dodecahedron Puzzle from the CFF:          Torsten Sillke, 15.09.96

  Contest 9 (reprinted in CFF 27, (1991) p5)

  Leo Links "inside the Icosahedron"

  Using the right number and the right combination of each
  of the four possible magnetic tetrahedral building blocks,
  we can construct a non-flipping model of the regular
  icosahedron.

  How many sets are possible and how many different
  arrangements exist for each set?

w, x, y, z count the number of tetrahedra with 0, 1, 2, or 3 black dots.
The digits 'a' to 'f' are standing for '10' to '15' (Hexadecimal Notation).
The symmetry-group of the icosahedron has not been factored out. In this
case its reflection-group S5 (order = 120) would be appropriate.
A further symmetry is the white/black exchange: solutions(wxyz) = solutions(zyxw).
It is better to analyze the problem by transforming to the dual polyhedron the
dodecahedron where all edge are oriented. For all 2^30 orientations I counted
the number of appearences of the different out-degree-seqences.

wxyz   solutions    Computation Time 16 minutes on a HP 755 (99 MHz).
----------------     The sum of all solutions is 2^30 = 1073741824.
0aa0    3600000
0b81   11280000
0c62    9846720
0d43    2778240
0e24     215040
0f05       2048
18b0   11280000
1991   51906560
1a72   68856960
1b53   31683840
1c34    4707840
1d15     153600
26c0    9846720
27a1   68856960
2882  140757120
2963  104117120
2a44   27541440
2b25    2108160
2c06      20480
34d0    2778240
35b1   31683840
3692  104117120
3773  125736960
3854   57803520
3935    8995840
3a16     314880
42e0     215040
43c1    4707840
44a2   27541440
4583   57803520
4664   47280000
4745   14240640
4826    1247040
4907      13440
50f0       2048
51d1     153600
52b2    2108160
5393    8995840
5474   14240640
5555    8414208
5636    1621120
5717      65280
60c2      20480
61a3     314880
6284    1247040
6365    1621120
6446     670080
6527      67200
6608        320
7094      13440
7175      65280
7256      67200
7337      15360
8066        320

Who counts (generates) the symmetric solutions.
As there are many symmetries many case have to be considered.

If the numbers are known for each case the symmetry group
can be factored out.