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%I #13 Mar 10 2024 13:26:25
%S 1,308,34128,1056576,15303750,136236276,865711763,4296782848,
%T 17656466751,62510672500,196174554026,557301826368,1456216515468,
%U 3543525156276,8109415963125,17592637669376,36414622551373
%N Number of achiral colorings of the 16 tetrahedral facets of a hyperoctahedron or of the 16 vertices of a tesseract.
%C An achiral coloring is identical to its reflection. The Schläfli symbols for the tesseract and the hyperoctahedron are {4,3,3} and {3,3,4} respectively. Both figures are regular 4-D polyhedra and they are mutually dual. There are 192 elements in the automorphism group of the tesseract that are not in its rotation group. Each involves a permutation of the axes that can be associated with a partition of 4 based on the conjugacy class of the permutation. This table shows the hyperoctahedron facet (tesseract vertex) cycle indices for each member of such a class. The first formula is obtained by averaging these cycle indices after replacing x_i^j with n^j according to the Pólya enumeration theorem.
%C Partition Count Odd Cycle Indices
%C 4 6 8x_1^2x_2^1x_4^3
%C 31 8 8x_2^2x_6^2
%C 22 3 8x_4^4
%C 211 6 2x_1^8x_2^4 + 2x_2^8 + 4x_4^4
%C 1111 1 8x_2^8
%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 a(n) = n^4 * (3*n^8 + 5*n^4 + 12*n^2 + 28) / 48.
%F a(n) = 1*C(n,1) + 306*C(n,2) + 33207*C(n,3) + 921908*C(n,4) + 10359075*C(n,5) + 59584470*C(n,6) + 197644440*C(n,7) + 400752240*C(n,8) + 505197000*C(n,9) + 386694000*C(n,10) + 164656800*C(n,11) + 29937600*C(n,12), where the coefficient of C(n,k) is the number of achiral colorings using exactly k colors.
%F a(n) = 2*A128767(n) - A337952(n) = A337952(n) - 2*A337954(n) = A128767(n) - A337954(n).
%t Table[(3n^12+5n^8+12n^6+28n^4)/48,{n,30}]
%Y Cf. A337952 (oriented), A128767 (unoriented), A337954 (chiral).
%Y Other elements: A331361 (tesseract edges, hyperoctahedron faces), A331357 (tesseract faces, hyperoctahedron edges), A337958 (tesseract facets, hyperoctahedron vertices).
%Y Other polychora: A132366(n-1) (4-simplex facets/vertices), A338951 (24-cell), A338967 (120-cell, 600-cell).
%Y Row 4 of A325015 (orthoplex facets, orthotope vertices).
%K nonn,easy
%O 1,2
%A _Robert A. Russell_, Oct 03 2020
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