A331354 -id:A331354 - cp's OEIS frontend

A331355 Number of unoriented colorings of the edges of a regular 4-dimensional orthoplex with n available colors.

1, 49127, 740360358, 733776248840, 155261523065875, 12340612271439081, 498926608780739307, 12298018390569089088, 207726683413584244680, 2604177120221402303875, 25650403577338260144611, 207023317470352041578712

Offset: 1

Robert A. Russell, Jan 14 2020

A regular 4-dimensional orthoplex (also hyperoctahedron or cross polytope) has 8 vertices and 24 edges. Its Schläfli symbol is {3,3,4}. Two unoriented colorings are the same if congruent; chiral pairs are counted as one. Also the number of unoriented colorings of the square faces of a tesseract {4,3,3} with n available colors.

Index entries for linear recurrences with constant coefficients, 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).

Cf. A331354 (oriented), A331356 (chiral), A331357 (achiral).
Other polychora: A063843 (5-cell), A331359 (8-cell), A338953 (24-cell), A338965 (120-cell, 600-cell).
Row 4 of A337412 (orthoplex edges, orthotope ridges) and A337888 (orthotope faces, orthoplex peaks).

Table[(48 n^3 + 64 n^4 + 44 n^6 + 84 n^7 + 56 n^8 + 12 n^9 + 5 n^12 +
    36 n^13 + 18 n^14 + 12 n^15 + 4 n^18 + n^24)/384, {n, 1, 25}]

a(n) = (48*n^3 + 64*n^4 + 44*n^6 + 84*n^7 + 56*n^8 + 12*n^9 + 5*n^12 +
   36*n^13 + 18*n^14 + 12*n^15 + 4*n^18 + n^24) / 384.
a(n) = C(n,1) + 49125*C(n, 2) + 740212980*C(n, 3) + 730815102166*C(n, 4) + 151600044933990*C(n, 5) + 11420034970306170*C(n, 6) + 415777158607920585*C(n, 7) + 8643499341510394200*C(n, 8) + 113988734942055623055*C(n, 9) + 1023002477284840979850*C(n, 10) + 6559265715033958749900*C(n, 11) + 31097943476763200314200*C(n, 12) + 111710751446923209781200*C(n, 13) + 309231173588248964052000*C(n, 14) + 666846649590586048584000*C(n, 15) + 1126625898539640346848000*C(n, 16) + 1492173541849975272288000*C(n, 17) + 1541987122059614438208000*C(n, 18) + 1229356526029003532160000*C(n, 19) + 741102367008078915840000*C(n, 20) + 326583680209195368960000*C(n, 21) + 99234043419574103040000*C(n, 22) + 18581137031073576960000*C(n, 23) + 1615751046180311040000*C(n, 24), where the coefficient of C(n,k) is the number of colorings using exactly k colors.
a(n) = A331354(n) - A331356(n) = (A331354(n) + A331357(n)) / 2 = A331356(n) + A331357(n).

A331356 Number of chiral pairs of colorings of the edges of a regular 4-dimensional orthoplex with n available colors.

Original entry on oeis.org

0, 40927, 731279799, 732272925320, 155180061396500, 12338466190481025, 498892380429882397, 12297640855782563904, 207723543409061974215, 2604156223742219218875, 25650287482426463967550, 207022761847763612943192

Offset: 1

Robert A. Russell, Jan 14 2020

A regular 4-dimensional orthoplex (also hyperoctahedron or cross polytope) has 8 vertices and 24 edges. Its Schläfli symbol is {3,3,4}. The chiral colorings of its edges come in pairs, each the reflection of the other. Also the number of chiral pairs of colorings of the square faces of a tesseract {4,3,3} with n available colors.

Index entries for linear recurrences with constant coefficients, 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).

Cf. A331354 (oriented), A331355 (unoriented), A331357 (achiral).
Other polychora: A331352 (5-cell), A331360 (8-cell), A338954 (24-cell), A338966 (120-cell, 600-cell).
Row 4 of A337413 (orthoplex edges, orthotope ridges) and A337889 (orthotope faces, orthoplex peaks).

Table[(48n^3 - 20n^6 - 60n^7 + 8n^8 + 12n^9 - 3n^12 + 12n^13 + 18n^14 - 12n^15 - 4n^18 + n^24)/384, {n, 1, 25}]

a(n) = (48*n^3 - 20*n^6 - 60*n^7 + 8*n^8 + 12*n^9 - 3*n^12 + 12*n^13 + 18*n^14 - 12*n^15 - 4*n^18 + n^24) / 384.
a(n) = 40927*C(n,2) + 731157018*C(n,3) + 729348051686*C(n,4) + 151526009158620*C(n,5) + 11418355290999750*C(n,6) + 415756294427389020*C(n,7) + 8643340000393019040*C(n,8) + 113987930725267657695*C(n,9) + 1022999668724320645050*C(n,10) + 6559258733377155798300*C(n,11) + 31097930936416379343000*C(n,12) + 111710735118080165667600*C(n,13) + 309231158315533166512800*C(n,14) + 666846639586795403736000*C(n,15) + 1126625894182469352672000*C(n,16) + 1492173540716221595232000*C(n,17) + 1541987121926231652672000*C(n,18) + 1229356526029003532160000*C(n,19) + 741102367008078915840000*C(n,20) + 326583680209195368960000*C(n,21) + 99234043419574103040000*C(n,22) + 18581137031073576960000*C(n,23) + 1615751046180311040000*C(n,24), where the coefficient of C(n,k) is the number of colorings using exactly k colors.
a(n) = A331354(n) - A331355(n) = (A331354(n) - A331357(n)) / 2 = A331355(n) - A331357(n).

A331357 Number of achiral colorings of the edges of a regular 4-dimensional orthoplex with n available colors.

Original entry on oeis.org

1, 8200, 9080559, 1503323520, 81461669375, 2146080958056, 34228350856910, 377534786525184, 3140004522270465, 20896479183085000, 116094911796177061, 555622588428635520, 2346039511676401359, 8903083257215729960

Offset: 1

Robert A. Russell, Jan 14 2020

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

G. Royle, Partitions and Permutations
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).

Cf. A331354 (oriented), A331355 (unoriented), A331356 (chiral).
Other polychora: A331353 (5-cell), A331361 (8-cell), A338955 (24-cell), A338967 (120-cell, 600-cell).
Row 4 of A337414 (orthoplex edges, orthotope ridges) and A337890 (orthotope faces, orthoplex peaks).

Table[(8n^4 + 8n^6 + 18n^7 + 6n^8 + n^12 + 3n^13 + 3n^15 + n^18)/48, {n, 1, 25}]

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.
a(n) = 2*A331355(n) - A331354(n) = A331354(n) - 2*A331356(n) = A331355(n) - A331356(n).

A331358 Number of oriented colorings of the edges of a tesseract with n available colors.

Original entry on oeis.org

1, 22409620, 9651199594275, 96076801068337216, 121265960728368199375, 41451359960612034644436, 5752227470227262715982165, 412646679764073090531066880, 17883769897375781105874361581

Offset: 1

Robert A. Russell, Jan 14 2020

A tesseract is a regular 4-dimensional orthotope or hypercube with 16 vertices and 32 edges. Its Schläfli symbol is {4,3,3}. 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 triangular faces of a regular 4-dimensional orthoplex {3,3,4} with n available colors.
There are 192 elements in the rotation group of the tesseract. 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  Even Cycle Indices
  4          6      8x_8^4
  31         8      4x_1^2x_3^10 + 4x_2^1x_6^5
  22         3      4x_2^16 + 4x_4^8
  211        6      4x_1^4x_2^14 + 4x_4^8
  1111       1      x_1^32 + 7x_2^16

G. Royle, Partitions and Permutations
Index entries for linear recurrences with constant coefficients, signature (33, -528, 5456, -40920, 237336, -1107568, 4272048, -13884156, 38567100, -92561040, 193536720, -354817320, 573166440, -818809200, 1037158320, -1166803110, 1166803110, -1037158320, 818809200, -573166440, 354817320, -193536720, 92561040, -38567100, 13884156, -4272048, 1107568, -237336, 40920, -5456, 528, -33, 1).

Cf. A331359 (unoriented), A331360 (chiral), A331361 (achiral).

Cf. A331350 (simplex), A331354 (orthoplex), A338952 (24-cell), A338964 (120-cell, 600-cell).

Table[(48n^4 + 32n^6 + 36n^8 + 32n^12 + 19n^16 + 24n^18 + n^32)/192, {n, 1, 25}]

a(n) = (48*n^4 + 32*n^6 + 36*n^8 + 32*n^12 + 19*n^16 + 24*n^18 + n^32) / 192.
a(n) = C(n,1) + 22409618*C(n,2) + 9651132365418*C(n,3) + 96038196404417832*C(n,4) + 120785673234798359850*C(n,5) + 40725205155234194765220*C(n,6) + 5464611173328028329053040*C(n,7) + 367782713912186945387883840*C(n,8) + 14373563321596798877701789800*C(n,9) + 359883141899402124632485810800*C(n,10) + 6184991837595074128351177096800*C(n,11) + 76711443861342809436413801659200*C(n,12) + 712777405284132776184971034460800*C(n,13) + 5104524541259652946568783959507200*C(n,14) + 28797485239301310151711610238720000*C(n,15) + 130163892496470993203014850790912000*C(n,16) + 477548461917280632356433595575936000*C(n,17) + 1436223810514558840121822575516416000*C(n,18) + 3566452148795758403208660387955200000*C(n,19) + 7348050481070906467554726390758400000*C(n,20) + 12594856495384277051085880584652800000*C(n,21) + 17969280084916069147800454551859200000*C(n,22) + 21302862405912312079825436975308800000*C(n,23) + 20896529603947922315711136828211200000*C(n,24) + 16837871283345549751877122560000000000*C(n,25) + 11021533432128296153318764634112000000*C(n,26) + 5764800913106992933428143603712000000*C(n,27) + 2351280741029830331492705206272000000*C(n,28) + 720354927933711780177833164800000000*C(n,29) + 155891316152123120086047129600000000*C(n,30) + 21242333189959633945791037440000000*C(n,31) + 1370473109029653802954260480000000*C(n,32), where the coefficient of C(n,k) is the number of colorings using exactly k colors.
a(n) = A331359(n) + A331360(n) = 2*A331359(n) - A331361(n) = 2*A331360(n) + A331361(n).

A331350 Number of oriented colorings of the edges (or triangular faces) of a regular 4-dimensional simplex with n available colors.

Original entry on oeis.org

1, 40, 1197, 18592, 166885, 1019880, 4738153, 17962624, 58248153, 166920040, 432738229, 1032709536, 2298857821, 4822806184, 9613704465, 18329410048, 33605960689, 59516325288, 102196242685, 170682720160, 278019522837

Offset: 1

Robert A. Russell, Jan 14 2020

A 4-dimensional simplex has 5 vertices and 10 edges. Its Schläfli symbol is {3,3,3}. Two oriented colorings are the same if one is a rotation of the other; chiral pairs are counted as two.
There are 60 elements in the rotation group of the 4-dimensional simplex. Each is an even 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  Even Cycle Indices
  5          24     x_5^2
  311        20     x_1^1x_3^3
  221        15     x_1^2x_2^4
  11111      1      x_1^10

G. Royle, Partitions and Permutations
Index entries for linear recurrences with constant coefficients, signature (11, -55, 165, -330, 462, -462, 330, -165, 55, -11, 1).

Cf. A063843 (unoriented), A331352 (chiral), A331353 (achiral).
Other polychora: A331358 (8-cell), A331354 (16-cell), A338952 (24-cell), A338964 (120-cell, 600-cell).
Row 4 of A327083 (simplex edges and facets) and A337883 (simplex faces and peaks).

Table[(24n^2 + 20n^4 + 15n^6 + n^10)/60, {n, 1, 25}]

a(n) = (24*n^2 + 20*n^4 + 15*n^6 + n^10) / 60.
a(n) = C(n,1) + 38*C(n,2) + 1080*C(n,3) + 14040*C(n,4) + 85500*C(n,5) + 274104*C(n,6) + 493920*C(n,7) + 504000*C(n,8) + 272160*C(n,9) + 60480*C(n,10), where the coefficient of C(n,k) is the number of colorings using exactly k colors.
a(n) = A063843(n) + A331352(n) = 2*A063843(n) - A331353(n) = 2*A331352(n) + A331353(n).

A337411 Array read by descending antidiagonals: T(n,k) is the number of oriented colorings of the edges of a regular n-dimensional orthoplex (cross polytope) using k or fewer colors.

Original entry on oeis.org

1, 2, 1, 3, 6, 1, 4, 24, 218, 1, 5, 70, 2285, 90054, 1, 6, 165, 703760, 1471640157, 573439556, 1, 7, 336, 10194250, 1466049174160, 6332134720430727, 50043770249328, 1, 8, 616, 90775566, 310441584462375, 629648890639384572032, 1839894096099964270283469, 59966884221697869216, 1

Offset: 1

Robert A. Russell, Aug 26 2020

Each chiral pair is counted as two when enumerating oriented arrangements. For n=1, the figure is a line segment with one edge. For n=2, the figure is a square with 4 edges. For n=3, the figure is an octahedron with 12 edges. The number of edges is 2n*(n-1) for n>1.

Also the number of oriented colorings of the regular (n-2)-dimensional orthotopes (hypercubes) in a regular n-dimensional orthotope.

			Table begins with T(1,1):
1   2     3      4        5        6         7          8           9 ...
1   6    24     70      165      336       616       1044        1665 ...
1 218 22815 703760 10194250 90775566 576941778 2863870080 11769161895 ...
For T(2,2)=6, the arrangements are AAAA, AAAB, AABB, ABAB, ABBB, and BBBB.

K. Balasubramanian, Computational enumeration of colorings of hyperplanes of hypercubes for all irreducible representations and applications, J. Math. Sci. & Mod. 1 (2018), 158-180.

Cf. A337412 (unoriented), A337413 (chiral), A337414 (achiral).
Rows 1-4 are A000027, A006528, A060530, A331354.
Cf. A327083 (simplex edges), A337407 (orthotope edges), A325004 (orthoplex vertices).

m=1; (* dimension of color element, here an edge *)
Fi1[p1_] := Module[{g, h}, Coefficient[Product[g = GCD[k1, p1]; h = GCD[2 k1, p1]; (1 + 2 x^(k1/g))^(r1[[k1]] g) If[Divisible[k1, h], 1, (1+2x^(2 k1/h))^(r2[[k1]] h/2)], {k1, Flatten[Position[cs, n1_ /; n1 > 0]]}], x, m+1]];
FiSum[] := (Do[Fi2[k2] = Fi1[k2], {k2, Divisors[per]}];DivisorSum[per, DivisorSum[d1 = #, MoebiusMu[d1/#] Fi2[#] &]/# &]);
CCPol[r_List] := (r1 = r; r2 = cs - r1; If[EvenQ[Sum[If[EvenQ[j3], r1[[j3]], r2[[j3]]], {j3,n}]], (per = LCM @@ Table[If[cs[[j2]] == r1[[j2]], If[0 == cs[[j2]],1,j2], 2j2], {j2,n}]; Times @@ Binomial[cs, r1] 2^(n-Total[cs]) b^FiSum[]),0]);
PartPol[p_List] := (cs = Count[p, #]&/@ Range[n]; Total[CCPol[#]&/@ Tuples[Range[0,cs]]]);
pc[p_List] := Module[{ci, mb}, mb = DeleteDuplicates[p]; ci = Count[p, #]&/@ mb; n!/(Times@@(ci!) Times@@(mb^ci))] (*partition count*)
row[m]=b;
row[n_Integer] := row[n] = Factor[(Total[(PartPol[#] pc[#])&/@ IntegerPartitions[n]])/(n! 2^(n-1))]
array[n_, k_] := row[n] /. b -> k
Table[array[n,d+m-n], {d,8}, {n,m,d+m-1}] // Flatten

The algorithm used in the Mathematica program below assigns each permutation of the axes to a partition of n and then considers separate conjugacy classes for axis reversals. It uses the formulas in Balasubramanian's paper. If the value of m is increased, one can enumerate colorings of higher-dimensional elements beginning with T(m,1).

T(n,k) = A337412(n,k) + A337413(n,k) = 2*A337412(n,k) - A337414(n,k) = 2*A337413(n,k) + A337414(n,k).

A338952 Number of oriented colorings of the 96 edges (or triangular faces) of the 4-D 24-cell using subsets of a set of n colors.

Original entry on oeis.org

1, 137548893254081168086800768, 11046328890861011039111168376671536861388643, 10897746068379654103881579020805286236644252743361724416

Offset: 1

Robert A. Russell, Nov 17 2020

Each chiral pair is counted as two when enumerating oriented arrangements. 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 rotation group of the 24-cell. They divide into 20 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   Even Cycle Indices      Count   Even Cycle Indices
       1   x_1^96              6+6+36+36   x_4^24
      72   x_1^4x_2^46                32   x_2^3x_6^15
    1+18   x_2^48                 8+8+32   x_6^16
      32   x_1^6x_3^30             72+72   x_8^12
  8+8+32   x_3^32                  48+48   x_12^8

Robert A. Russell, Table of n, a(n) for n = 1..30
Index entries for linear recurrences with constant coefficients, order 97.

Cf. A338953 (unoriented), A338954 (chiral), A338955 (achiral), A338956 (exactly n colors), A338948 (vertices, facets), A331350 (5-cell), A331358 (8-cell edges, 16-cell faces), A331354 (16-cell edges, 8-cell faces), A338964 (120-cell, 600-cell).

Table[(96n^8+144n^12+48n^16+32n^18+84n^24+48n^32+32n^36+19n^48+72n^50+n^96)/576,{n,15}]

a(n) = (96*n^8 + 144*n^12 + 48*n^16 + 32*n^18 + 84*n^24 + 48*n^32 + 32*n^36 + 19*n^48 + 72*n^50 + n^96) / 576.
a(n) = Sum_{j=1..Min(n,96)} A338956(n) * binomial(n,j).
a(n) = A338953(n) + A338954(n) = 2*A338953(n) - A338955(n) = 2*A338954(n) + A338955(n).

A337952 Number of oriented colorings of the 16 tetrahedral facets of a hyperoctahedron or of the 16 vertices of a tesseract.

Original entry on oeis.org

1, 496, 230076, 22456756, 795467350, 14697611496, 173107727191, 1466088119056, 9651378868011, 52083991149400, 239323201136866, 962942859342036, 3465720389989936, 11343525530430016, 34210497067620525

Offset: 1

Robert A. Russell, Oct 03 2020

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
        6      8x_8^2
       8      4x_1^4x_3^4 + 4x_2^2x_6^2
       3      4x_1^4x_2^6 + 4x_4^4
      6      4x_2^8 + 4x_4^4
     1      x_1^16 + 7x_2^8

Index entries for linear recurrences with constant coefficients, signature (17, -136, 680, -2380, 6188, -12376, 19448, -24310, 24310, -19448, 12376, -6188, 2380, -680, 136, -17, 1).

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).

Table[(n^16+12n^10+63n^8+68n^4+48n^2)/192,{n,30}]

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).

A337956 Number of oriented colorings of the 8 cubic facets of a tesseract or of the 8 vertices of a hyperoctahedron.

Original entry on oeis.org

1, 15, 126, 730, 3270, 11991, 37450, 102726, 253485, 573265, 1205556, 2384460, 4475926, 8031765, 13858860, 23106196, 37372545, 58837851, 90421570, 135971430, 200486286, 290376955, 413769126, 580852650, 804281725

Offset: 1

Robert A. Russell, Oct 03 2020

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.

Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A337957 (unoriented), A234249(n+1) (chiral), A337958 (achiral).
Other elements: A331354 (hyperoctahedron edges, tesseract faces), A331358 (hyperoctahedron faces, tesseract edges), A337952 (hyperoctahedron facets, tesseract vertices).
Other polychora: A337895 (5-cell), A338948 (24-cell), A338964 (120-cell, 600-cell).
Row 4 of A325004 (orthotope facets, orthoplex vertices).

Table[Binomial[Binomial[n+1,2]+3,4] + Binomial[Binomial[n,2],4],{n,30}]

a(n) = binomial(binomial(n+1,2)+3,4) + binomial(binomial(n,2),4).
a(n) = n * (n+1) * (n^6 - n^5 + 7*n^4 + 29*n^3 + 16*n^2 - 4*n + 48) / 192.
a(n) = 1*C(n,1) + 13*C(n,2) + 84*C(n,3) + 312*C(n,4) + 735*C(n,5) + 1020*C(n,6) + 735*C(n,7) + 210*C(n,8), where the coefficient of C(n,k) is the number of oriented colorings using exactly k colors.
a(n) = A337957(n) + A234249(n+1) = 2*A337957(n) - A337958(n) = 2*A234249(n+1) + A337958(n).
From Stefano Spezia, Oct 04 2020: (Start)
G.f.: x*(1 + 6*x + 27*x^2 + 52*x^3 + 102*x^4 + 21*x^5 + x^6)/(1 - x)^9.
a(n) = 9*a(n-1)-36*a(n-2)+84*a(n-3)-126*a(n-4)+126*a(n-5)-84*a(n-6)+36*a(n-7)-9*a(n-8)+a(n-8) for n > 8.
(End)

A337887 Array read by descending antidiagonals: T(n,k) is the number of oriented colorings of the square faces of a regular n-dimensional orthotope (hypercube) using k or fewer colors.

Original entry on oeis.org

1, 2, 1, 3, 10, 1, 4, 57, 90054, 1, 5, 240, 1471640157, 629648865588086369152, 1, 6, 800, 1466049174160, 76983765319971901895960429658208179, 76686070519895153193719509580895099970955878067526648007224125292544, 1

Offset: 2

Robert A. Russell, Sep 28 2020

Each chiral pair is counted as two when enumerating oriented arrangements. Each face is a square bounded by four edges. For n=2, the figure is a square with one face. For n=3, the figure is a cube with 6 faces. For n=4, the figure is a tesseract with 24 faces. The number of faces is 2^(n-2)*C(n,2).

Also the number of oriented colorings of peaks of an n-dimensional orthoplex. A peak is an (n-3)-dimensional simplex.

			Array begins with T(2,1):
 1     2          3             4               5                 6 ...
 1    10         57           240             800              2226 ...
 1 90054 1471640157 1466049174160 310441584462375 24679078461920106 ...

K. Balasubramanian, Computational enumeration of colorings of hyperplanes of hypercubes for all irreducible representations and applications, J. Math. Sci. & Mod. 1 (2018), 158-180.

Cf. A337888 (unoriented), A337889 (chiral), A337890 (achiral).
Other elements: A325012 (vertices), A337407 (edges).
Other polytopes: A337883 (simplex), A337891 (orthoplex).
Rows 2-4 are A000027, A047780, A331354.

m = 2;(* dimension of color element, here a square face *)
Fi1[p1_] := Module[{g, h}, Coefficient[Product[g = GCD[k1, p1]; h = GCD[2 k1, p1]; (1 + 2 x^(k1/g))^(r1[[k1]] g) If[Divisible[k1, h], 1, (1+2x^(2 k1/h))^(r2[[k1]] h/2)], {k1, Flatten[Position[cs, n1_ /; n1 > 0]]}], x, n - m]];
FiSum[] := (Do[Fi2[k2] = Fi1[k2], {k2, Divisors[per]}];DivisorSum[per, DivisorSum[d1 = #, MoebiusMu[d1/#] Fi2[#] &]/# &]);
CCPol[r_List] := (r1 = r; r2 = cs - r1; If[EvenQ[Sum[If[EvenQ[j3], r1[[j3]], r2[[j3]]], {j3,n}]], (per = LCM @@ Table[If[cs[[j2]] == r1[[j2]], If[0 == cs[[j2]],1,j2], 2j2], {j2,n}]; Times @@ Binomial[cs, r1] 2^(n-Total[cs]) b^FiSum[]),0]);
PartPol[p_List] := (cs = Count[p, #]&/@ Range[n]; Total[CCPol[#]&/@ Tuples[Range[0,cs]]]);
pc[p_List] := Module[{ci, mb}, mb = DeleteDuplicates[p]; ci = Count[p, #]&/@ mb; n!/(Times@@(ci!) Times@@(mb^ci))] (*partition count*)
row[n_Integer] := row[n] = Factor[(Total[(PartPol[#] pc[#])&/@ IntegerPartitions[n]])/(n! 2^(n-1))]
array[n_, k_] := row[n] /. b -> k
Table[array[n,d+m-n], {d,6}, {n,m,d+m-1}] // Flatten

The algorithm used in the Mathematica program below assigns each permutation of the axes to a partition of n and then considers separate conjugacy classes for axis reversals. It uses the formulas in Balasubramanian's paper. If the value of m is increased, one can enumerate colorings of higher-dimensional elements beginning with T(m,1).

T(n,k) = A337888(n,k) + A337889(n,k) = 2*A337888(n,k) - A337890(n,k) = 2*A337889(n,k) + A337890(n,k).

cp's OEIS Frontend

A331355 Number of unoriented colorings of the edges of a regular 4-dimensional orthoplex with n available colors.

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A331356 Number of chiral pairs of colorings of the edges of a regular 4-dimensional orthoplex with n available colors.

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A331357 Number of achiral colorings of the edges of a regular 4-dimensional orthoplex with n available colors.

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A331358 Number of oriented colorings of the edges of a tesseract with n available colors.

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A331350 Number of oriented colorings of the edges (or triangular faces) of a regular 4-dimensional simplex with n available colors.

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A337411 Array read by descending antidiagonals: T(n,k) is the number of oriented colorings of the edges of a regular n-dimensional orthoplex (cross polytope) using k or fewer colors.

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A338952 Number of oriented colorings of the 96 edges (or triangular faces) of the 4-D 24-cell using subsets of a set of n colors.

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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.

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