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%I #15 Mar 10 2024 13:24:20
%S 1,496,230076,22456756,795467350,14697611496,173107727191,
%T 1466088119056,9651378868011,52083991149400,239323201136866,
%U 962942859342036,3465720389989936,11343525530430016,34210497067620525
%N Number of oriented colorings of the 16 tetrahedral facets of a hyperoctahedron or of the 16 vertices of a tesseract.
%C Each chiral pair is counted as two when enumerating oriented arrangements. 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 rotation group of the tesseract. 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 cycle indices for each rotation by partition. 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 Even Cycle Indices
%C 4 6 8x_8^2
%C 31 8 4x_1^4x_3^4 + 4x_2^2x_6^2
%C 22 3 4x_1^4x_2^6 + 4x_4^4
%C 211 6 4x_2^8 + 4x_4^4
%C 1111 1 x_1^16 + 7x_2^8
%H <a href="/index/Rec#order_17">Index entries for linear recurrences with constant coefficients</a>, signature (17, -136, 680, -2380, 6188, -12376, 19448, -24310, 24310, -19448, 12376, -6188, 2380, -680, 136, -17, 1).
%F a(n) = n^2 * (n^14 + 12*n^8 + 63*n^6 + 68*n^2 + 48) / 192.
%F a(n) = 1*C(n,1) + 494*C(n,2) + 228591*C(n,3) + 21539424*C(n,4) + 685479375*C(n,5) + 10257064650*C(n,6) + 86151316860*C(n,7) + 449772354360*C(n,8) + 1551283253100*C(n,9) + 3661969537800*C(n,10) + 6015983173200*C(n,11) + 6878457986400*C(n,12) + 5371454088000*C(n,13) + 2733402672000*C(n,14) + 817296480000*C(n,15) + 108972864000*C(n,16), where the coefficient of C(n,k) is the number of oriented colorings using exactly k colors.
%F a(n) = A128767(n) + A337954(n) = 2*A128767(n) - A337955(n) = 2*A337954(n) + A337955(n).
%t Table[(n^16+12n^10+63n^8+68n^4+48n^2)/192,{n,30}]
%Y Cf. A128767 (unoriented), A337954 (chiral), A337955 (achiral).
%Y Other elements: A331358 (tesseract edges, hyperoctahedron faces), A331354 (tesseract faces, hyperoctahedron edges), A337956 (tesseract facets, hyperoctahedron vertices).
%Y Other polychora: A337895 (4-simplex facets/vertices), A338948 (24-cell), A338964 (120-cell, 600-cell).
%Y Row 4 of A325012 (orthoplex facets, orthotope vertices).
%K nonn,easy
%O 1,2
%A _Robert A. Russell_, Oct 03 2020
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