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A338948 Number of oriented colorings of the 24 octahedral facets (or 24 vertices) of the 4-D 24-cell using subsets of a set of n colors.

%I A338948 #14 Mar 10 2024 13:34:13
%S A338948 1,30968,490710246,488689596200,103480643539150,8226360697111116,
%T A338948 332606338581801018,8198553131754111456,138483409168412322525,
%U A338948 1736111115543474313600,17100230356306262961356,138015359782116886130568
%N A338948 Number of oriented colorings of the 24 octahedral facets (or 24 vertices) of the 4-D 24-cell using subsets of a set of n colors.
%C A338948 Each chiral pair is counted as two when enumerating oriented arrangements. The Schläfli symbol of the 24-cell is {3,4,3}. It is self-dual. There are 576 elements in the rotation group of the 24-cell. They divide into 20 conjugacy classes. The first formula is obtained by averaging the vertex (or facet) cycle indices after replacing x_i^j with n^j according to the Pólya enumeration theorem.
%C A338948    Count   Even Cycle Indices      Count   Even Cycle Indices
%C A338948        1   x_1^24                     36   x_2^2x_4^5
%C A338948       18   x_1^4x_2^10                32   x_2^3x_6^3
%C A338948       72   x_1^2x_2^11               6+6   x_4^6
%C A338948        1   x_2^12                 8+8+32   x_6^4
%C A338948       32   x_1^6x_3^6              72+72   x_8^3
%C A338948       36   x_1^4x_4^5              48+48   x_12^2
%C A338948   8+8+32   x_3^8
%H A338948 Robert A. Russell, <a href="/A338948/b338948.txt">Table of n, a(n) for n = 1..30</a>
%H A338948 <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 A338948 a(n) = (96*n^2 + 144*n^3 + 48*n^4 + 44*n^6 + 36*n^7 + 48*n^8 + 36*n^9 + 33*n^12 + 72*n^13 + 18*n^14 + n^24) / 576.
%F A338948 a(n) = 1*C(n,1) + 30966*C(n,2) + 490617345*C(n,3) + 486726941020*C(n,4) + 101042102350935*C(n,5) + 7612797366078810*C(n,6) + 277177820254686645*C(n,7) + 5762279787373449480*C(n,8) + 75992221900428179850*C(n,9) + 682000715348622816300*C(n,10) + 4372841482811937689400*C(n,11) + 20731958137729666674000*C(n,12) + 74473828855001644068000*C(n,13) + 206154110634594043521600*C(n,14) + 444564429725793817440000*C(n,15) + 751083930907369899840000*C(n,16) + 994782360855398955840000*C(n,17) + 1027991414661948696960000*C(n,18) + 819571017352669021440000*C(n,19) + 494068244672052610560000*C(n,20) + 217722453472796912640000*C(n,21) + 66156028946382735360000*C(n,22) + 12387424687382384640000*C(n,23) + 1077167364120207360000*C(n,24), where the coefficient of C(n,k) is the number of oriented colorings using exactly k colors.
%F A338948 a(n) = A338949(n) + A338950(n) = 2*A338949(n) - A338951(n) = 2*A338950(n) + A338951(n).
%t A338948 Table[(96n^2+144n^3+48n^4+44n^6+36n^7+48n^8+36n^9+33n^12+72n^13+18n^14+n^24)/576,{n,15}]
%Y A338948 Cf. A338949 (unoriented), A338950 (chiral), A338951 (achiral), A338952 (edges, faces), A337895 (5-cell), A337952 (8-cell vertices, 16-cell facets), A337956 (16-cell vertices, 8-cell facets), A338964 (120-cell, 600-cell).
%K A338948 nonn,easy
%O A338948 1,2
%A A338948 _Robert A. Russell_, Nov 17 2020