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 A337952 Number of oriented colorings of the 16 tetrahedral facets of a hyperoctahedron or of the 16 vertices of a tesseract. 8
 1, 496, 230076, 22456756, 795467350, 14697611496, 173107727191, 1466088119056, 9651378868011, 52083991149400, 239323201136866, 962942859342036, 3465720389989936, 11343525530430016, 34210497067620525 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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.   Partition  Count  Even Cycle Indices   4          6      8x_8^2   31         8      4x_1^4x_3^4 + 4x_2^2x_6^2   22         3      4x_1^4x_2^6 + 4x_4^4   211        6      4x_2^8 + 4x_4^4   1111       1      x_1^16 + 7x_2^8 LINKS FORMULA a(n) = n^2 * (n^14 + 12*n^8 + 63*n^6 + 68*n^2 + 48) / 192. 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. a(n) = A128767(n) + A337954(n) = 2*A128767(n) - A337955(n) = 2*A337954(n) + A337955(n). MATHEMATICA Table[(n^16+12n^10+63n^8+68n^4+48n^2)/192, {n, 30}] CROSSREFS Cf. A128767 (unoriented), A337954 (chiral), A337955 (achiral). Other elements: A331358 (tesseract edges, hyperoctahedron faces), A331354 (tesseract faces, hyperoctahedron edges), A337956 (tesseract facets, hyperoctahedron vertices). Other polychora: A337895 (4-simplex facets/vertices), A338948 (24-cell), A338964 (120-cell, 600-cell). Row 4 of A325012 (orthoplex facets, orthotope vertices). Sequence in context: A110847 A160472 A028512 * A088845 A013685 A108157 Adjacent sequences:  A337949 A337950 A337951 * A337953 A337954 A337955 KEYWORD nonn,easy AUTHOR Robert A. Russell, Oct 03 2020 STATUS approved

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Last modified December 8 19:21 EST 2021. Contains 349596 sequences. (Running on oeis4.)