cp's OEIS Frontend

A060530 Number of inequivalent ways to color edges of a cube using at most n colors.

%I A060530 #32 Nov 28 2024 11:13:02
%S A060530 0,1,218,22815,703760,10194250,90775566,576941778,2863870080,
%T A060530 11769161895,41669295250,130772947481,371513523888,970769847320,
%U A060530 2362273657030,5406141568500,11728193258496,24276032182173,48201464902410,92221684354915
%N A060530 Number of inequivalent ways to color edges of a cube using at most n colors.
%C A060530 Here inequivalent means under the action of the rotation group of the cube, of order 24, which in its action on the edges has cycle index (x1^12 + 3*x2^6 + 6*x4^3 + 6*x1^2*x2^5 + 8*x3^4)/24.
%C A060530 Also, number of inequivalent colorings of the edges of a regular octahedron using at most n colors. - _José H. Nieto S._, Jan 19 2012
%C A060530 From _Robert A. Russell_, Oct 08 2020: (Start)
%C A060530 Each chiral pair is counted as two when enumerating oriented arrangements. The Schläfli symbols for the regular octahedron and cube are {3,4} and {4,3} respectively. They are mutually dual.
%C A060530 There are 24 elements in the rotation group of the regular octahedron/cube. They divide into five conjugacy classes. The first formula is obtained by averaging the edge cycle indices after replacing x_i^j with n^j according to the Pólya enumeration theorem.
%C A060530   Conjugacy Class    Count    Even Cycle Indices
%C A060530   Identity              1     x_1^12
%C A060530   Vertex rotation       8     x_3^4
%C A060530   Edge rotation         6     x_1^2x_2^5
%C A060530   Small face rotation   6     x_4^3
%C A060530   Large face rotation   3     x_2^6 (End)
%C A060530 Also, number of ways of coloring the vertices of the truncated tetrahedron or faces of the triakis tetrahedron up to rotation and reflection. - _Peter Kagey_, Nov 27 2024
%D A060530 N. G. De Bruijn, Polya's theory of counting, in E. F. Beckenbach, ed., Applied Combinatorial Mathematics, Wiley, 1964, pp. 144-184 (see p. 147).
%H A060530 Harry J. Smith, <a href="/A060530/b060530.txt">Table of n, a(n) for n=0..200</a>
%H A060530 <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 A060530 a(n) = (n^12 + 6*n^7 + 3*n^6 + 8*n^4 + 6*n^3)/24. (Replace all x_i's in the cycle index by n.)
%F A060530 G.f.: -x*(150*x^10 +19758*x^9 +425032*x^8 +2763481*x^7 +6769435*x^6 +6773089*x^5 +2763307*x^4 +423883*x^3 +20059*x^2 +205*x +1)/(x -1)^13. - _Colin Barker_, Aug 13 2012
%F A060530 From _Robert A. Russell_, Oct 08 2020: (Start)
%F A060530 a(n) = 1*C(n,1) + 216*C(n,2) + 22164*C(n,3) + 613804*C(n,4) + 6901425*C(n,5) + 39713430*C(n,6) + 131754420*C(n,7) + 267165360*C(n,8) + 336798000*C(n,9) + 257796000*C(n,10) + 109771200*C(n,11) + 19958400*C(n,12), where the coefficient of C(n,k) is the number of oriented colorings using exactly k colors.
%F A060530 a(n) = A199406(n) + A337406(n) = 2*A199406(n) - A331351(n) = 2*A337406(n) + A331351(n). (End)
%t A060530 Table[(n^12+6n^7+3n^6+8n^4+6n^3)/24,{n,0,20}] (* _Harvey P. Dale_, Feb 13 2013 *)
%o A060530 (PARI) { for (n=0, 200, write("b060530.txt", n, " ", (n^12 + 6*n^7 + 3*n^6 + 8*n^4 + 6*n^3)/24); ) } \\ _Harry J. Smith_, Jul 06 2009
%Y A060530 Cf. A199406 (unoriented), A337406 (chiral), A331351 (achiral).
%Y A060530 Other elements: A000543 (cube vertices, octahedron faces), A047780 (cube faces, octahedron vertices).
%Y A060530 Cf. A046023 (tetrahedron), A282670 (dodecahedron/icosahedron).
%Y A060530 Row 3 of A337407 (orthotope edges, orthoplex ridges) and A337411 (orthoplex edges, orthotope ridges).
%K A060530 nonn,easy
%O A060530 0,3
%A A060530 _N. J. A. Sloane_, Apr 11 2001
%E A060530 Entry revised by _N. J. A. Sloane_, Jan 03 2005