cp's OEIS Frontend

A331351 Number of achiral colorings of the edges of a cube or regular octahedron.

%I A331351 #32 Mar 04 2024 00:46:43
%S A331351 1,70,1407,12480,69050,281946,931490,2632512,6598935,15041950,
%T A331351 31740841,62830560,117855192,211141490,363551700,604679936,975561405,
%U A331351 1531968822,2348375395,3522668800,5181705606,7487800650,10646250902
%N A331351 Number of achiral colorings of the edges of a cube or regular octahedron.
%C A331351 A cube has 8 vertices and 12 edges. A regular octahedron has 6 vertices and 12 edges. An achiral coloring is identical to its reflection.
%C A331351 From _Robert A. Russell_, Oct 08 2020: (Start)
%C A331351 The Schläfli symbols for the cube and regular octahedron are {4,3} and {3,4} respectively. They are mutually dual.
%C A331351 There are 24 elements in the automorphism group of the regular octahedron/cube that are not in the rotation group. 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 A331351   Conjugacy Class     Count    Odd Cycle Indices
%C A331351   Inversion              1     x_2^6
%C A331351   Vertex rotation*       8     x_6^2            Asterisk indicates that the
%C A331351   Edge rotation*         6     x_1^2x_2^5       operation is followed by an
%C A331351   Small face rotation*   3     x_4^3            inversion.
%C A331351   Large face rotation*   6     x_1^4x_2^4 (End)
%H A331351 G. Royle, <a href="http://teaching.csse.uwa.edu.au/units/CITS7209/partition.pdf">Partitions and Permutations</a>
%H A331351 <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 A331351 a(n) = (8*n^2 + 6*n^3 + n^6 + 6*n^7 + 3*n^8) / 24.
%F A331351 a(n) = 1*C(n,1) + 68*C(n,2) + 1200*C(n,3) + 7268*C(n,4) + 20025*C(n,5) + 27750*C(n,6) + 18900*C(n,7) + 5040*C(n,8), where the coefficient of C(n,k) is the number of colorings using exactly k colors.
%F A331351 a(n) = 2*A199406(n) - A060530(n) = A060530(n) - 2*A337406(n) = A199406(n) - A337406(n). - _Robert A. Russell_, Oct 08 2020
%F A331351 G.f.: (x + 61*x^2 + 813*x^3 + 2253*x^4 + 1628*x^5 + 282*x^6 + 2*x^7) / (1-x)^9.
%F A331351 E.g.f.: (1/24)*exp(x)*x*(24 + 816*x + 4800*x^2 + 7268*x^3 + 4005*x^4 + 925*x^5 + 90*x^6 + 3*x^7). - _Stefano Spezia_, Jan 17 2020
%t A331351 Table[(8n^2 + 6n^3 + n^6 + 6n^7 + 3n^8)/24, {n, 1, 30}]
%t A331351 LinearRecurrence[{9, -36, 84, -126, 126, -84, 36, -9, 1}, {1, 70, 1407, 12480, 69050, 281946, 931490, 2632512, 6598935}, 25]
%Y A331351 Cf. A060530 (oriented), A199406 (unoriented), A337406 (chiral), A337897 (octahedron faces, cube vertices), A337898 (cube faces, octahedron vertices), A037270 (tetrahedron), A337953 (dodecahedron, icosahedron).
%Y A331351 Row 3 of A337410 (orthotope edges, orthoplex ridges) and A337414 (orthoplex edges, orthotope ridges).
%K A331351 nonn,easy
%O A331351 1,2
%A A331351 _Robert A. Russell_, Jan 14 2020