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%I #11 Mar 09 2024 12:22:48
%S 1,93024,294157089,91983927296,7960001890625,304914963625056,
%T 6652124939544609,96100248309858304,1013293206632601441,
%U 8334166666733500000,56066328722011832961,319495406392484665344
%N Number of achiral colorings of the edges of a tesseract with n available colors.
%C A tesseract is a regular 4-dimensional orthotope or hypercube with 16 vertices and 32 edges. Its Schläfli symbol is {4,3,3}. An achiral coloring is identical to its reflection. Also the number of achiral colorings of the triangular faces of a regular 4-dimensional orthoplex {3,3,4} with n available colors.
%C There are 192 elements in the automorphism group of the tesseract that are not in its rotation group. Each is associated with a partition of 4 based on the conjugacy group of the permutation of the axes. The first formula is obtained by averaging their 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_4^8
%C 31 8 4x_1^2x_3^2x_6^4 + 4x_2^1x_6^5
%C 22 3 8x_4^8
%C 211 6 2x_1^8x_2^12 + 2x_2^16 + 4x_4^8
%C 1111 1 4x_1^8x_2^12 + 4x_2^16
%H G. Royle, <a href="http://teaching.csse.uwa.edu.au/units/CITS7209/partition.pdf">Partitions and Permutations</a>
%H <a href="/index/Rec#order_21">Index entries for linear recurrences with constant coefficients</a>, signature (21, -210, 1330, -5985, 20349, -54264, 116280, -203490, 293930, -352716, 352716, -293930, 203490, -116280, 54264, -20349, 5985, -1330, 210, -21, 1).
%F a(n) = (2*n^6 + 8*n^8 + n^16 + n^20) / 12.
%F a(n) = C(n,1) + 93022*C(n,2) + 293878020*C(n,3) + 90807857080*C(n,4) + 7503022894800*C(n,5) + 258528829444320*C(n,6) + 4681671089961600*C(n,7) + 50981530073846400*C(n,8) + 363246007692204000*C(n,9) + 1789536284820648000*C(n,10) + 6323058513173001600*C(n,11) + 16406578807069651200*C(n,12) + 31689737477798400000*C(n,13) + 45786987328642560000*C(n,14) + 49291621471572480000*C(n,15) + 38970361271761920000*C(n,16) + 21972146261345280000*C(n,17) + 8363100653107200000*C(n,18) + 1926047423139840000*C(n,19) + 202741834014720000*C(n,20), where the coefficient of C(n,k) is the number of colorings using exactly k colors.
%F a(n) = 2*A331359(n) - A331358(n) = A331358(n) - 2*A331360(n) = A331359(n) - A331360(n).
%t Table[(2n^6 + 8n^8 + n^16 + n^20)/12, {n, 1, 25}]
%Y Cf. A331358 (oriented), A331359 (unoriented), A331360 (chiral).
%Y Cf. A331353 (simplex), A331357 (orthoplex), A338955 (24-cell), A338967 (120-cell, 600-cell).
%K nonn,easy
%O 1,2
%A _Robert A. Russell_, Jan 14 2020
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