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%I #16 Mar 09 2024 12:00:28
%S 1,90054,1471640157,1466049174160,310441584462375,24679078461920106,
%T 997818989210621704,24595659246351652992,415450226822646218895,
%U 5208333343963621522750,51300691059764724112161,414046079318115654521904
%N Number of oriented colorings of the edges of a regular 4-dimensional orthoplex with n available colors.
%C A regular 4-dimensional orthoplex (also hyperoctahedron or cross polytope) has 8 vertices and 24 edges. Its Schläfli symbol is {3,3,4}. Two oriented colorings are the same if one is a rotation of the other; chiral pairs are counted as two. Also the number of oriented colorings of the square faces of a tesseract {4,3,3} with n available colors.
%C There are 192 elements in the rotation group of the 4-dimensional orthoplex. 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 Even Cycle Indices
%C 4 6 8x_8^3
%C 31 8 4x_3^8 + 4x_6^4
%C 22 3 4x_1^4x_2^10 + 4x_4^6
%C 211 6 4x_1^2x_2^11 + 2x_1^4x_4^5 + 2x_2^2x_4^5
%C 1111 1 6x_1^4x_2^10 + x_1^24 + x_2^12
%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_25">Index entries for linear recurrences with constant coefficients</a>, signature (25, -300, 2300, -12650, 53130, -177100, 480700, -1081575, 2042975, -3268760, 4457400, -5200300, 5200300, -4457400, 3268760, -2042975, 1081575, -480700, 177100, -53130, 12650, -2300, 300, -25, 1).
%F a(n) = (48*n^3 + 32*n^4 + 12*n^6 + 12*n^7 + 32*n^8 + 12*n^9 + n^12 + 24*n^13 + 18*n^14 + n^24) / 192.
%F a(n) = C(n,1) + 90052*C(n,2) + 1471369998*C(n,3) + 1460163153852*C(n,4) + 303126054092610*C(n,5) + 22838390261305920*C(n,6) + 831533453035309605*(n,7) + 17286839341903413240*C(n,8) + 227976665667323280750*C(n,9) + 2046002146009161624900*C(n,10) + 13118524448411114548200*C(n,11) + 62195874413179579657200*C(n,12) + 223421486565003375448800*C(n,13) + 618462331903782130564800*C(n,14) + 1333693289177381452320000*C(n,15) + 2253251792722109699520000*C(n,16) + 2984347082566196867520000*C(n,17) + 3083974243985846090880000*C(n,18) + 2458713052058007064320000*C(n,19) + 1482204734016157831680000*C(n,20) + 653167360418390737920000*C(n,21) + 198468086839148206080000*C(n,22) + 37162274062147153920000*C(n,23) + 3231502092360622080000*C(n,24), where the coefficient of C(n,k) is the number of colorings using exactly k colors.
%F a(n) = A331355(n) + A331356(n) = 2*A331355(n) - A331357(n) = 2*A331356(n) + A331357(n).
%t Table[(48n^3 + 32n^4 + 12n^6 + 12n^7 + 32n^8 + 12n^9 + n^12 + 24n^13 + 18n^14 + n^24)/192, {n, 1, 25}]
%Y Cf. A331355 (unoriented), A331356 (chiral), A331357 (achiral).
%Y Other polychora: A331350 (5-cell), A331358 (8-cell), A338952 (24-cell), A338964 (120-cell, 600-cell).
%Y Row 4 of A337411 (orthoplex edges, orthotope ridges) and A337887 (orthotope faces, orthoplex peaks).
%K nonn,easy
%O 1,2
%A _Robert A. Russell_, Jan 14 2020
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