|
%I #46 Jun 24 2023 21:39:46
%S 1,96,9099,280832,4073375,36292320,230719293,1145393152,4707296613,
%T 16666924000,52307593239,148602435840,388302646355,944900450144,
%U 2162441849625,4691253854208,9710376716137,19280531603808,36888593841475,68266682784000,122597146773927
%N Number of ways of n-coloring a dodecahedron.
%C More explicitly, a(n) is the number of colorings with at most n colors of the faces of a regular dodecahedron, inequivalent under the action of the rotation group of the dodecahedron. It is also the number of inequivalent colorings of the vertices of a regular icosahedron using at most n colors. - _José H. Nieto S._, Jan 19 2012
%C From _Robert A. Russell_, Oct 03 2020: (Start)
%C Each chiral pair is counted as two when enumerating oriented arrangements. The Schläfli symbols for the regular icosahedron and regular dodecahedron are {3,5} and {5,3} respectively. They are mutually dual. There are 60 elements in the rotation group of the regular dodecahedron/icosahedron. They divide into five conjugacy classes. The first formula is obtained by averaging the dodecahedron face (icosahedron vertex) cycle indices after replacing x_i^j with n^j according to the Pólya enumeration theorem.
%C Conjugacy Class Count Even Cycle Indices
%C Identity 1 x_1^12
%C Edge rotation 15 x_2^6
%C Vertex rotation 20 x_3^4
%C Small face rotation 12 x_1^2x_5^2
%C Large face rotation 12 x_1^2x_5^2 (End)
%H T. D. Noe, <a href="/A000545/b000545.txt">Table of n, a(n) for n = 1..1000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PolyhedronColoring.html">Polyhedron Coloring</a>
%H <a href="/index/Rec#order_13">Index entries for linear recurrences with constant coefficients</a>, signature (13, -78, 286, -715, 1287, -1716, 1716, -1287, 715, -286, 78, -13, 1).
%F G.f.: x*((1+x)*(1+x*(82+x*(7847+x*(161900+x*(943640+x*(1764740+x*(943640+x*(161900+x*(7847+x*(82+x)))))))))))/(1-x)^13. - _Harvey P. Dale_, Apr 25 2011
%F From _Robert A. Russell_, Oct 03 2020: (Start)
%F a(n) = (n^12 + 15*n^6 + 44*n^4) / 60.
%F a(n) = 1*C(n,1) + 94*C(n,2) + 8814*C(n,3) + 245008*C(n,4) + 2759250*C(n,5) + 15884004*C(n,6) + 52701264*C(n,7) + 106866144*C(n,8) + 134719200*C(n,9) + 103118400*C(n,10) + 43908480*C(n,11) + 7983360*C(n,12), where the coefficient of C(n,k) is the number of oriented colorings using exactly k colors.
%F a(n) = A252705(n) + A337961(n) = 2*A252705(n) - A337962(n) = 2*A337961(n) + A337962(n). (End)
%p (1/60)*n^12+(1/4)*n^6+(11/15)*n^4;
%t Table[n^12/60+n^6/4+11 n^4/15,{n,20}] (* or *) CoefficientList[Series[ -(((1+x) (1+x (82+x (7847+x (161900+x (943640+x (1764740+x (943640+x (161900+x (7847+x (82+x)))))))))))/(x-1)^13),{x,0,20}],x] (* _Harvey P. Dale_, Apr 25 2011 *)
%Y Cf. A252705 (unoriented), A337961 (chiral), A337962 (achiral).
%Y Other elements: A054472 (dodecahedron vertices, icosahedron faces), A282670 (edges).
%Y Other polyhedra: A006008 (tetrahedron), A047780 (cube faces, octahedron vertices), A000543 (octahedron faces, cube vertices).
%K nonn,easy
%O 1,2
%A Clint. C. Williams (Clintwill(AT)aol.com)
| 