A000543 - OEIS

%I #43 Sep 08 2022 08:44:28

%S 0,1,23,333,2916,16725,70911,241913,701968,1798281,4173775,8942021,

%T 17930628,34009053,61518471,106823025,179003456,290715793,459239463,

%U 707740861,1066780100,1576090341,2286660783,3263156073,4586706576

%N Number of inequivalent ways to color vertices of a cube using at most n colors.

%C Here inequivalent means under the action of the rotation group of the cube, of order 24, which in its action on the vertices has cycle index (x1^8 + 9*x2^4 + 6*x4^2 + 8*x1^2*x3^2)/24.

%C Also the number of ways to color the faces of a regular octahedron with n colors, counting mirror images separately.

%C From _Robert A. Russell_, Oct 08 2020: (Start)

%C 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 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 cube vertex (octahedron face) 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_1^8

%C   Vertex rotation       8     x_1^2x_3^2

%C   Edge rotation         6     x_2^4

%C   Small face rotation   6     x_4^2

%C   Large face rotation   3     x_2^4 (End)

%D 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 Vincenzo Librandi, <a href="/A000543/b000543.txt">Table of n, a(n) for n = 0..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_09">Index entries for linear recurrences with constant coefficients</a>, signature (9,-36,84,-126,126,-84,36,-9,1).

%F a(n) = (1/24)*n^2*(n^6+17*n^2+6). (Replace all x_i's in the cycle index with n.)

%F G.f.: x*(1+x)*(1+13*x+149*x^2+514*x^3+149*x^4+13*x^5+x^6)/(1-x)^9. - _Colin Barker_, Jan 29 2012

%F a(n) = 1*C(n,1) + 21*C(n,2) + 267*C(n,3) + 1718*C(n,4) + 5250*C(n,5) + 7980*C(n,6) + 5880*C(n,7) + 1680*C(n,8), where the coefficient of C(n,k) is the number of oriented colorings using exactly k colors.

%F a(n) = A128766(n) + A337896(n) = 2*A128766(n) - A337897(n) = 2*A337896(n) + A337897(n). - _Robert A. Russell_, Oct 08 2020

%p f:= n->(1/24)*n^2*(n^6+17*n^2+6); seq(f(n), n=0..40);

%t CoefficientList[Series[x*(1+x)*(1+13*x+149*x^2+514*x^3+149*x^4+13*x^5+x^6)/(1-x)^9,{x,0,30}],x] (* _Vincenzo Librandi_, Apr 15 2012 *)

%t Table[(n^8+17n^4+6n^2)/24,{n,0,30}] (* _Robert A. Russell_, Oct 08 2020 *)

%o (Magma) [(1/24)*n^2*(n^6+17*n^2+6): n in [0..30]]; // _Vincenzo Librandi_, Apr 15 2012

%Y Cf. A128766 (unoriented), A337896 (chiral), A337897 (achiral).

%Y Other elements: A060530 (edges), A047780 (cube faces, octahedron vertices).

%Y Cf. A006008 (tetrahedron), A000545 (dodecahedron faces, icosahedron vertices), A054472 (icosahedron faces, dodecahedron vertices).

%Y Row 3 of A325012 (orthotope vertices, orthoplex facets) and A337891 (orthoplex faces, orthotope peaks).

%K nonn,easy

%O 0,3

%A Clint. C. Williams (Clintwill(AT)aol.com)

%E Entry revised by _N. J. A. Sloane_, Jan 03 2005