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.