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%I #5 Sep 08 2022 08:45:14
%S 5,10,23,96,17824
%N Number of different colorings of the platonic solids allowing rotation symmetry.
%C Two colorings of a platonic solid are said to be the same if one is able to pick up the solid and rotate it in such a way as to align the colors.
%o (Magma) // Tetraeder S4 := SymmetricGroup( 4 ); r := S4 ! (2,3,4); s := S4 ! (1,2)(3,4); tetraeder := sub< S4 | r, s >; // Hexaeder S6 := SymmetricGroup( 6 ); r := S6 ! (2,3,4,5); s := S6 ! (1,3,4)(2,6,5); hexaeder := sub< S6 | r, s >; // Octaeder S8 := SymmetricGroup( 8 ); r := S8 ! (1,2,3,4)(5,6,7,8); s := S8 ! (1,2,6,5)(3,7,8,4); octaeder := sub< S8 | r, s >; // Dodecaeder S12 := SymmetricGroup( 12 ); r := S12 ! (2,3,4,5,6)(7,8,9,10,11); s := S12 ! (1,3,7,11,6)(4,8,12,10,5); dodecaeder := sub< S12 | r, s >; // Icosaeder S20 := SymmetricGroup( 20 ); r := S20 ! (1,2,3,4,5)(6,8,10,12,14)(7,9,11,13,15)(16,17,18,19,20); s := S20 ! (1,2,8,7,6)(3,9,16,15,5)(10,17,20,14,4)(11,18,19,13,12); icosaeder := sub< S20 | r, s >; for G in [tetraeder, hexaeder, octaeder, dodecaeder, icosaeder] do &+[ c[2] * n^( &+[ t[2]: t in CycleStructure( c[3] ) ] ): c in C ] / #G; end for;
%K nonn,fini,full
%O 1,1
%A Daan Wanrooy (wanrooy(AT)math.ru.nl), Sep 24 2004
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