Displaying 1-10 of 12 results found.
Number of oriented colorings of the 120 dodecahedral facets of the 4-D 120-cell (or 120 vertices of the 4-D 600-cell) using subsets of a set of n colors.
+10
13
1, 184614999414571937405905419562272, 249584763877004334779608333505026056531601345365910986, 245395425663664490219902430658740012166428009430164733569180712873472
COMMENTS
Each chiral pair is counted as two when enumerating oriented arrangements. The Schläfli symbols of the 120-cell and 600-cell are {5,3,3} and {3,3,5} respectively. They are mutually dual. There are 7200 elements in the rotation group of the 120-cell. They divide into 41 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.
Count Even Cycle Indices Count Even Cycle Indices
1 x_1^120 400 x_2^3x_6^19
450 x_1^4x_2^58 20+20 x_6^20
1 x_2^60 144+144 x_2^5x_10^11
400 x_1^6x_3^38 4*12+2*144 x_10^12
20+20 x_3^40 600+600 x_12^10
144+144 x_1^10x_5^22 4*240 x_15^8
30+30 x_4^30 4*360 x_20^6
4*12+2*144 x_5^24 4*240 x_30^4
Sequences for other elements of the 120-cell and 600-cell are not suitable for the OEIS as the first significant datum is too big. We provide formulas here.
For the 600 facets of the 600-cell (vertices of the 120-cell), the cycle indices are:
Count Even Cycle Indices Count Even Cycle Indices
1 x_1^600 400 x_2^6x_6^98
450 x_1^4x_2^298 20+20 x_6^100
1 x_2^300 4*12+4*144 x_10^60
400 x_1^12x_3^196 600+600 x_12^50
20+20 x_3^200 4*240 x_15^40
30+30 x_4^150 4*360 x_20^30
4*12+4*144 x_5^120 4*240 x_30^20
The formula is (960*n^20 + 1440*n^30 + 960*n^40 + 1200*n^50 + 624*n^60 + 40*n^100 + 400*n^104 + 624*n^120 + 60*n^150 + 40*n^200 + 400*n^208 + n^300 + 450*n^302 + n^600) / 7200.
For the 720 pentagonal faces of the 120-cell (edges of the 600-cell), the cycle indices are:
Count Even Cycle Indices Count Even Cycle Indices
1 x_1^720 2*20+400 x_6^120
450 x_1^8x_2^356 144+144 x_2^5x_10^71
1 x_2^360 4*12+2*144 x_10^72
2*20+400 x_3^240 600+600 x_12^60
30+30 x_4^180 4*240 x_15^48
144+144 x_1^10x_5^142 4*360 x_20^36
4*12+2*144 x_5^144 4*240 x_30^24
The formula is (960*n^24 + 1440*n^36 + 960*n^48 + 1200*n^60 + 336*n^72 + 288*n^76 + 440*n^120 + 336*n^144 + 288*n^152 + 60*n^180 + 440*n^240 + n^360 + 450*n^364 + n^720) / 7200.
For the 1200 edges of the 120-cell (triangular faces of the 600-cell), the cycle indices are:
Count Even Cycle Indices Count Even Cycle Indices
1 x_1^1200 400 x_2^3x_6^199
450 x_1^8x_2^596 20+20 x_6^200
1 x_2^600 4*12+4*144 x_10^120
400 x_1^6x_3^398 600+600 x_12^100
20+20 x_3^400 4*240 x_15^80
30+30 x_4^300 4*360 x_20^60
4*12+4*144 x_5^240 4*240 x_30^40
The formula is (960*n^40 + 1440*n^60 + 960*n^80 + 1200*n^100 + 624*n^120 + 40*n^200 + 400*n^202 + 624*n^240 + 60*n^300 + 40*n^400 + 400*n^404 + n^600 + 450*n^604 + n^1200) / 7200.
FORMULA
a(n) = (960*n^4 + 1440*n^6 + 960*n^8 + 1200*n^10 + 336*n^12 + 288*n^16 + 40*n^20 + 400*n^22 + 336*n^24 + 60* n^30 + 288*n^32 + 40*n^40 + 400*n^44 + n^60 + 450*n^62 + n^120) / 7200.
a(n) = Sum_{j=1..Min(n,120)} A338980(n) * binomial(n,j).
MATHEMATICA
Table[(960n^4+1440n^6+960n^8+1200n^10+336n^12+288n^16+40n^20+400n^22+336n^24+60n^30+288n^32+40n^40+400n^44 +n^60+450n^62 +n^120)/7200, {n, 10}]
PROG
(PARI) a(n)=(960*n^4+1440*n^6+960*n^8+1200*n^10+336*n^12+288*n^16+40*n^20+400*n^22+336 *n^24+60*n^30+288*n^32+40*n^40+400*n^44+n^60+450*n^62+n^120)/7200 \\ Charles R Greathouse IV, Jul 05 2024
Number of unoriented colorings of the 120 dodecahedral facets of the 4-D 120-cell (or 120 vertices of the 4-D 600-cell) using subsets of a set of n colors.
+10
13
1, 92307499707443390526727850063504, 124792381938502167392338612231208163827413085862945471, 122697712831832245109951221276235414511846772206539032522116543043328
COMMENTS
Each chiral pair is counted as one when enumerating unoriented arrangements. The Schläfli symbols of the 120-cell and 600-cell are {5,3,3} and {3,3,5} respectively. They are mutually dual.
Sequences for other elements of the 120-cell and 600-cell are not suitable for the OEIS as the first significant datum is too big. We provide formulas here.
For the 600 facets of the 600-cell (vertices of the 120-cell), the formula is (960*n^20 + 1440*n^30 + 960*n^40 + 1200*n^50 + 2064*n^60 + 1440*n^66 + 40*n^100 + 1600*n^104 + 1200*n^114 + 624*n^120 + 60*n^150 + 1800*n^152 + 40*n^200 + 400*n^208 + 61*n^300 + 450*n^302 + 60*n^330 + n^600) / 14400.
For the 720 pentagonal faces of the 120-cell (edges of the 600-cell), the formula is (960 n^24 + 1440 n^36 + 960 n^48 + 1200 n^60 + 336 n^72 + 1728 n^76 + 1440 n^84 + 1640 n^120 + 1200 n^132 + 336 n^144 + 288 n^152 + 60 n^180 + 1800 n^182 + 440 n^240 + 61 n^360 + 450 n^364 + 60 n^396 + n^720) / 14400.
For the 1200 edges of the 120-cell (triangular faces of the 600-cell), the formula is (960*n^40 + 1440*n^60 + 960*n^80 + 1200*n^100 + 2064*n^120 + 1440*n^128 + 40*n^200 + 1600*n^202 + 1200*n^216 + 624*n^240 + 60*n^300 + 1800*n^302 + 40*n^400 + 400*n^404 + 61*n^600 + 450*n^604 + 60*n^640 + n^1200) / 14400.
FORMULA
a(n) = (960*n^4 + 1440*n^6 + 960*n^8 + 1200*n^10 + 336*n^12 + 288*n^16 + 1440*n^17 + 1440*n^19 + 40*n^20 + 400*n^22 + 1200*n^23 + 336*n^24 + 1200*n^27 + 60*n^30 + 1800*n^31 + 288*n^32 + 40*n^40 + 400*n^44 + n^60 + 60*n^61 + 450*n^62 + 60*n^75 +*n^120) / 14400.
a(n) = Sum_{j=1..Min(n,120)} A338981(n) * binomial(n,j).
MATHEMATICA
Table[(960n^4+1440n^6+960n^8+1200n^10+336n^12+288n^16+1440n^17+1440n^19+40n^20+400n^22+1200n^23+336n^24+1200n^27+60n^30+1800n^31+288n^32+40n^40+400n^44+n^60+60n^61+450n^62+60n^75+n^120)/14400, {n, 10}]
Number of chiral pairs of colorings of the 120 dodecahedral facets of the 4-D 120-cell (or 120 vertices of the 4-D 600-cell) using subsets of a set of n colors.
+10
13
92307499707128546879177569498768, 124792381938502167387269721273817892704188259502965515, 122697712831832245109951209382504597654581237223625701047064169830144
COMMENTS
Each member of a chiral pair is a reflection but not a rotation of the other. The Schläfli symbols of the 120-cell and 600-cell are {5,3,3} and {3,3,5} respectively. They are mutually dual.
Sequences for other elements of the 120-cell and 600-cell are not suitable for the OEIS as the first significant datum is too big. We provide formulas here.
For the 600 facets of the 600-cell (vertices of the 120-cell), the formula is (960*n^20 + 1440*n^30 + 960*n^40 + 1200*n^50 - 816*n^60 - 1440*n^66 + 40*n^100 - 800*n^104 - 1200*n^114 + 624*n^120 + 60*n^150 - 1800*n^152 + 40*n^200 + 400*n^208 - 59*n^300 + 450*n^302 - 60*n^330 + n^600) / 14400.
For the 720 pentagonal faces of the 120-cell (edges of the 600-cell), the formula is (960 n^24 + 1440 n^36 + 960 n^48 + 1200 n^60 + 336 n^72 - 1152 n^76 - 1440 n^84 - 760 n^120 - 1200 n^132 + 336 n^144 + 288 n^152 + 60 n^180 - 1800 n^182 + 440 n^240 - 59 n^360 + 450 n^364 - 60 n^396 + n^720) / 14400.
For the 1200 edges of the 120-cell (triangular faces of the 600-cell), the formula is (960*n^40 + 1440*n^60 + 960*n^80 + 1200*n^100 - 816*n^120 - 1440*n^128 + 40*n^200 - 800*n^202 - 1200*n^216 + 624*n^240 + 60*n^300 - 1800*n^302 + 40*n^400 + 400*n^404 - 59*n^600 + 450*n^604 - 60*n^640 + n^1200) / 14400.
FORMULA
a(n) = (960*n^4 + 1440*n^6 + 960*n^8 + 1200*n^10 + 336*n^12 + 288*n^16 - 1440*n^17 - 1440*n^19 + 40*n^20 + 400*n^22 - 1200*n^23 + 336*n^24 - 1200*n^27 + 60*n^30 - 1800*n^31 + 288*n^32 + 40*n^40 + 400*n^44 + n^60 - 60*n^61 + 450*n^62 - 60*n^75 + n^120) / 14400.
a(n) = Sum_{j=2..Min(n,120)} A338982(n) * binomial(n,j).
MATHEMATICA
Table[(960n^4 +1440n^6 +960n^8 +1200n^10 +336n^12 +288n^16 -1440n^17 -1440n^19 +40n^20 +400n^22 -1200n^23 +336n^24 -1200n^27 +60n^30 -1800n^31 +288n^32 +40n^40 +400n^44 +n^60 -60n^61 +450n^62 -60n^75 +n^120)/14400, {n, 2, 10}]
Partial sum of centered tetrahedral numbers A005894.
+10
12
1, 6, 21, 56, 125, 246, 441, 736, 1161, 1750, 2541, 3576, 4901, 6566, 8625, 11136, 14161, 17766, 22021, 27000, 32781, 39446, 47081, 55776, 65625, 76726, 89181, 103096, 118581, 135750, 154721, 175616, 198561, 223686, 251125, 281016, 313501, 348726, 386841
COMMENTS
a(n-1) is the number of achiral colorings of the 5 tetrahedral facets (or vertices) of a regular 4-dimensional simplex using n or fewer colors. An achiral arrangement is identical to its reflection. The 4-dimensional simplex is also called a 5-cell or pentachoron. Its Schläfli symbol is {3,3,3}.
There are 60 elements in the automorphism group of the 4-dimensional simplex that are not in its rotation group. Each is an odd permutation of the vertices and can be associated with a partition of 5 based on the conjugacy class of the permutation. The first formula for a(n-1) is obtained by averaging their cycle indices after replacing x_i^j with n^j according to the Pólya enumeration theorem.
Partition Count Odd Cycle Indices
41 30 x_1x_4^1
32 20 x_2^1x_3^1
2111 10 x_1^3x_2^1 (End)
FORMULA
a(n) = (n^4 + 4*n^3 + 11*n^2 + 14*n + 6)/6.
a(n-1) = n^2 * (5 + n^2) / 6.
a(n-1) = binomial(n+4,5) - binomial(n,5).
a(n-1) = 1*C(n,1) + 4*C(n,2) + 6*C(n,3) + 4*C(n,4), where the coefficient of C(n,k) is the number of achiral colorings using exactly k colors.
G.f. for a(n-1): x * (x+1) * (x^2+1) / (1-x)^5. (End)
Sum_{n>=0} 1/a(n) = Pi^2/5 + 3/25 - 3*Pi*coth(sqrt(5)*Pi)/(5*sqrt(5)).
Sum_{n>=0} (-1)^n/a(n) = Pi^2/10 - 3/25 + 3*Pi*cosech(sqrt(5)*Pi)/(5*sqrt(5)). (End)
MATHEMATICA
Do[Print[n, " ", (n^4 + 4 n^3 + 11 n^2 + 14 n + 6)/6 ], {n, 0, 10000}]
Accumulate[Table[(2n+1)(n^2+n+3)/3, {n, 0, 40}]] (* or *) LinearRecurrence[ {5, -10, 10, -5, 1}, {1, 6, 21, 56, 125}, 40] (* Harvey P. Dale, Feb 26 2020 *)
CROSSREFS
Cf. A000292, A005894, A063488, A001845, A063489, A005898, A063490, A057813, A063491, A005902, A063492, A005917, A063493, A063494, A063495, A063496. Cf. A337895 (oriented), A000389(n+4) (unoriented), A000389 (chiral), A331353 (5-cell edges, faces), A337955 (8-cell vertices, 16-cell facets), A337958 (16-cell vertices, 8-cell facets), A338951 (24-cell), A338967 (120-cell, 600-cell).
EXTENSIONS
Corrected offset, Mathematica program by Tomas J. Bulka (tbulka(AT)rodincoil.com), Sep 02 2009
Number of achiral colorings of the edges (or triangular faces) of a regular 4-dimensional simplex with n available colors.
+10
11
1, 28, 387, 2784, 13125, 46836, 137543, 349952, 797769, 1667500, 3248971, 5973408, 10459917, 17571204, 28479375, 44742656, 68393873, 102041532, 148984339, 213340000, 300189141, 415735188, 567481047, 764423424
COMMENTS
A 4-dimensional simplex has 5 vertices and 10 edges. Its Schläfli symbol is {3,3,3}. An achiral coloring is identical to its reflection,
There are 60 elements in the automorphism group of the 4-dimensional simplex that are not in its rotation group. Each is an odd permutation of the vertices and can be associated with a partition of 5 based on the conjugacy group of the permutation. The first formula is obtained by averaging their cycle indices after replacing x_i^j with n^j according to the Pólya enumeration theorem.
Partition Count Odd Cycle Indices
41 30 x_2^1x_4^2
32 20 x_1^1x_3^1x_6^1
2111 10 x_1^4x_2^3
FORMULA
a(n) = (5*n^3 + n^7) / 6.
a(n) = C(n,1) + 26*C(n,2) + 306*C(n,3) + 1400*C(n,4) + 2800*C(n,5) + 2520*C(n,6) + 840*C(n,7), where the coefficient of C(n,k) is the number of colorings using exactly k colors.
G.f.: x*(1 + 20*x + 191*x^2 + 416*x^3 + 191*x^4 + 20*x^5 + x^6) / (1 - x)^8.
a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8) for n>8.
(End)
MATHEMATICA
Table[(5 n^3 + n^7)/6, {n, 1, 25}]
PROG
(PARI) Vec(x*(1 + 20*x + 191*x^2 + 416*x^3 + 191*x^4 + 20*x^5 + x^6) / (1 - x)^8 + O(x^25)) \\ Colin Barker, Jan 15 2020
CROSSREFS
Row 4 of A327086 (simplex edges and ridges) and A337886 (simplex faces and peaks).
Number of achiral colorings of the edges of a regular 4-dimensional orthoplex with n available colors.
+10
11
1, 8200, 9080559, 1503323520, 81461669375, 2146080958056, 34228350856910, 377534786525184, 3140004522270465, 20896479183085000, 116094911796177061, 555622588428635520, 2346039511676401359, 8903083257215729960
COMMENTS
A regular 4-dimensional orthoplex (also hyperoctahedron or cross polytope) has 8 vertices and 24 edges. Its Schläfli symbol is {3,3,4}. An achiral coloring is identical to its reflection. Also the number of achiral colorings of the square faces of a tesseract {4,3,3} with n available colors.
There are 192 elements in the automorphism group of the 4-dimensional orthoplex 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.
Partition Count Odd Cycle Indices
4 6 8x_2^2x_4^5
31 8 4x_3^4x_6^2 + 4x_6^4
22 3 8x_1^2x_2^1x_4^5
211 6 2x_1^2x_2^11 + 2x_1^6x_2^9 + 4x_2^2x_4^5
1111 1 4x_1^12x_2^6 + 4x_2^12
LINKS
Index entries for linear recurrences with constant coefficients, signature (19, -171, 969, -3876, 11628, -27132, 50388, -75582, 92378, -92378, 75582, -50388, 27132, -11628, 3876, -969, 171, -19, 1).
FORMULA
a(n) = (8*n^4 + 8*n^6 + 18*n^7 + 6*n^8 + n^12 + 3*n^13 + 3*n^15 + n^18) / 48.
a(n) = C(n,1) + 8198*C(n,2) + 9055962*C(n,3) + 1467050480*C(n,4) + 74035775370*C(n,5) + 1679679306420*C(n,6) + 20864180531565*C(n,7) + 159341117375160*C(n,8) + 804216787965360*C(n,9) + 2808560520334800*C(n,10) + 6981656802951600*C(n,11) + 12540346820971200*C(n,12) + 16328843044113600*C(n,13) + 15272715797539200*C(n,14) + 10003790644848000*C(n,15) + 4357170994176000*C(n,16) + 1133753677056000*C(n,17) + 133382785536000*C(n,18), where the coefficient of C(n,k) is the number of colorings using exactly k colors.
MATHEMATICA
Table[(8n^4 + 8n^6 + 18n^7 + 6n^8 + n^12 + 3n^13 + 3n^15 + n^18)/48, {n, 1, 25}]
CROSSREFS
Row 4 of A337414 (orthoplex edges, orthotope ridges) and A337890 (orthotope faces, orthoplex peaks).
Number of achiral colorings of the edges of a tesseract with n available colors.
+10
11
1, 93024, 294157089, 91983927296, 7960001890625, 304914963625056, 6652124939544609, 96100248309858304, 1013293206632601441, 8334166666733500000, 56066328722011832961, 319495406392484665344
COMMENTS
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.
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.
Partition Count Odd Cycle Indices
4 6 8x_4^8
31 8 4x_1^2x_3^2x_6^4 + 4x_2^1x_6^5
22 3 8x_4^8
211 6 2x_1^8x_2^12 + 2x_2^16 + 4x_4^8
1111 1 4x_1^8x_2^12 + 4x_2^16
LINKS
Index entries for linear recurrences with constant coefficients, signature (21, -210, 1330, -5985, 20349, -54264, 116280, -203490, 293930, -352716, 352716, -293930, 203490, -116280, 54264, -20349, 5985, -1330, 210, -21, 1).
FORMULA
a(n) = (2*n^6 + 8*n^8 + n^16 + n^20) / 12.
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.
MATHEMATICA
Table[(2n^6 + 8n^8 + n^16 + n^20)/12, {n, 1, 25}]
Number of achiral colorings of the 24 octahedral facets (or 24 vertices) of the 4-D 24-cell using subsets of a set of n colors.
+10
11
1, 6504, 8416440, 1455789440, 80139247500, 2125945744776, 34026498820524, 376045864704000, 3131319814422255, 20854395850585000, 115919421344402676, 554976171149122944, 2343894146343268610, 8896568181794053320
COMMENTS
An achiral coloring is identical to its reflection. The Schläfli symbol of the 24-cell is {3,4,3}. It is self-dual. There are 576 elements in the automorphism group of the 24-cell that are not in its rotation group. They divide into 10 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.
Count Odd Cycle Indices Count Odd Cycle Indices
12 x_1^12x_2^6 72 x_2^2x_4^5
12 x_1^6x_2^9 96 x_1^2x_2^2x_6^3
12 x_1^2x_2^11 96 x_2^3x_3^2x_6^2
12 x_2^12 96 x_3^4x_6^2
72 x_1^2x_2^1x_4^5 96 x_6^4
LINKS
Index entries for linear recurrences with constant coefficients, signature (19, -171, 969, -3876, 11628, -27132, 50388, -75582, 92378, -92378, 75582, -50388, 27132, -11628, 3876, -969, 171, -19, 1).
FORMULA
a(n) = (8*n^4 + 8*n^6 + 22*n^7 + 6*n^8 + n^12 + n^13 + n^15 + n^18) / 48.
a(n) = 1*C(n,1) + 6502*C(n,2) + 8396931*C(n,3) + 1422162700*C(n,4) + 72944399665*C(n,5) + 1666778870130*C(n,6) + 20777144613015*C(n,7) + 158973991255800*C(n,8) + 803196369526320*C(n,9) + 2806639981714800*C(n,10) + 6979192091902800*C(n,11) + 12538220293368000*C(n,12) + 16327662245294400*C(n,13) + 15272334392515200*C(n,14) + 10003736158416000*C(n,15) + 4357170994176000*C(n,16) + 1133753677056000*C(n,17) + 133382785536000*C(n,18), where the coefficient of C(n,k) is the number of achiral colorings using exactly k colors.
MATHEMATICA
Table[(8n^4+8n^6+22n^7+6n^8+n^12+n^13+n^15+n^18)/48, {n, 15}]
CROSSREFS
Cf. A338948 (oriented), A338949 (unoriented), A338950 (chiral), A338955 (edges, faces), A132366 (5-cell), A337955 (8-cell vertices, 16-cell facets), A337958 (16-cell vertices, 8-cell facets), A338967 (120-cell, 600-cell).
Number of achiral colorings of the 96 edges (or triangular faces) of the 4-D 24-cell using subsets of a set of n colors.
+10
9
1, 24124751133507584, 883287060208158070437496209, 27692675763559261523047959805034496, 18070082615414169898334284655914306640625, 1018202231744161700740376040914469837333037056
COMMENTS
An achiral coloring is identical to its reflection. The Schläfli symbol of the 24-cell is {3,4,3}. It has 24 octahedral facets. It is self-dual. There are 576 elements in the automorphism group of the 24-cell that are not in its rotation group. They divide into 10 conjugacy classes. The first formula is obtained by averaging the edge (or face) cycle indices after replacing x_i^j with n^j according to the Pólya enumeration theorem.
Count Odd Cycle Indices Count Odd Cycle Indices
12 x_1^24x_2^36 96 x_1^2x_2^2x_3^2x_6^14
12 x_1^8x_2^44 96 x_3^8x_6^12
12+12 x_3^48 96 x_2^3x_6^15
72+72 x_4^24 96 x_6^16
FORMULA
a(n) = (8*n^16 + 8*n^18 + 16*n^20 + 12*n^24 + 2*n^48 + n^52 + n^60) / 48.
a(n) = Sum_{j=1..Min(n,60)} A338959(n) * binomial(n,j).
MATHEMATICA
Table[(8n^16+8n^18+16n^20+12n^24+2n^48+n^52+n^60)/48, {n, 15}]
CROSSREFS
Cf. A338952 (oriented), A338953 (unoriented), A338954 (chiral), A338959 (exactly n colors), A338951 (vertices, facets), A331353 (5-cell), A331361 (8-cell edges, 16-cell faces), A331357 (16-cell edges, 8-cell faces), A338967 (120-cell, 600-cell).
Number of achiral colorings of the 16 tetrahedral facets of a hyperoctahedron or of the 16 vertices of a tesseract.
+10
8
1, 308, 34128, 1056576, 15303750, 136236276, 865711763, 4296782848, 17656466751, 62510672500, 196174554026, 557301826368, 1456216515468, 3543525156276, 8109415963125, 17592637669376, 36414622551373
COMMENTS
An achiral coloring is identical to its reflection. 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 automorphism group of the tesseract that are not in its rotation group. 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 hyperoctahedron facet (tesseract vertex) cycle indices for each member of such a class. 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 Odd Cycle Indices
4 6 8x_1^2x_2^1x_4^3
31 8 8x_2^2x_6^2
22 3 8x_4^4
211 6 2x_1^8x_2^4 + 2x_2^8 + 4x_4^4
1111 1 8x_2^8
LINKS
Index entries for linear recurrences with constant coefficients, signature (13, -78, 286, -715, 1287, -1716, 1716, -1287, 715, -286, 78, -13, 1).
FORMULA
a(n) = n^4 * (3*n^8 + 5*n^4 + 12*n^2 + 28) / 48.
a(n) = 1*C(n,1) + 306*C(n,2) + 33207*C(n,3) + 921908*C(n,4) + 10359075*C(n,5) + 59584470*C(n,6) + 197644440*C(n,7) + 400752240*C(n,8) + 505197000*C(n,9) + 386694000*C(n,10) + 164656800*C(n,11) + 29937600*C(n,12), where the coefficient of C(n,k) is the number of achiral colorings using exactly k colors.
MATHEMATICA
Table[(3n^12+5n^8+12n^6+28n^4)/48, {n, 30}]
CROSSREFS
Other elements: A331361 (tesseract edges, hyperoctahedron faces), A331357 (tesseract faces, hyperoctahedron edges), A337958 (tesseract facets, hyperoctahedron vertices). Other polychora: A132366(n-1) (4-simplex facets/vertices), A338951 (24-cell), A338967 (120-cell, 600-cell). Row 4 of A325015 (orthoplex facets, orthotope vertices).
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