A331355 -id:A331355 - cp's OEIS frontend

A063843 Number of n-multigraphs on 5 nodes.

0, 1, 34, 792, 10688, 90005, 533358, 2437848, 9156288, 29522961, 84293770, 217993600, 519341472, 1154658869, 2420188694, 4821091920, 9187076352, 16837177281, 29809183410, 51172613512, 85448030080, 139159855989, 221554769150, 345523218536, 528767663040

Offset: 0

N. J. A. Sloane, Aug 25 2001

Equivalently, number of ways to color edges of complete graph on 5 nodes with n colors, under action of symmetric group S_5, of order 120, with cycle index on edges given by (1/120)*(24*x5^2 + 30*x2*x4^2 + 20*x3^3*x1 + 20*x3*x6*x1 + 15*x1^2*x2^4 + 10*x1^4*x2^3 + x1^10). Setting all x_i = n gives the sequence.
Number of vertex colorings of the Petersen graph. Marko Riedel, Mar 24 2016
Number of unoriented colorings of the 10 triangular edges or triangular faces of a pentachoron, Schläfli symbol {3,3,3}, using n or fewer colors. Also called a 5-cell or 4-simplex. - Robert A. Russell, Oct 17 2020

T. D. Noe, Table of n, a(n) for n = 0..1000
Vladeta Jovovic, Formulae for the number T(n,k) of n-multigraphs on k nodes
Marko Riedel, Vertex colorings of the Petersen graph, Math StackExchange
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Cf. A063842. A row of A063841.
Cf. A331350 (oriented), A331352 (chiral), A331353 (achiral), A000389(n+4) (vertices and facets)
Other polychora: A331359 (8-cell), A331355 (16-cell), A338953 (24-cell), A338965 (120-cell, 600-cell).
Row 4 of A327084 (simplex edges and ridges) and A337884 (simplex faces and peaks).

f:=n-> 1/120*(24*n^2+50*n^3+20*n^4+15*n^6+10*n^7+n^10);

Table[(24n^2+50n^3+20n^4+15n^6+10n^7+n^10)/120,{n,0,30}] (* or *) LinearRecurrence[{11,-55,165,-330,462,-462,330,-165,55,-11,1},{0,1,34,792,10688,90005,533358,2437848,9156288,29522961,84293770},30] (* Harvey P. Dale, Oct 20 2012 *)

a(n)=n^2*(n^8+10*n^5+15*n^4+20*n^2+50*n+24)/120 \\ Charles R Greathouse IV, Jan 20 2012

a(n) = (1/120)*(24*n^2+50*n^3+20*n^4+15*n^6+10*n^7+n^10).
a(n+1) = (1/5!)*(n^10 + 10*n^9 + 45*n^8 + 130*n^7 + 295*n^6 + 552*n^5 + 805*n^4 + 900*n^3 + 774*n^2 + 448*n + 120).
G.f. = (1 + 23*x + 473*x^2 + 3681*x^3 + 10717*x^4 + 11221*x^5 + 3779*x^6 + 339*x^7 + 6*x^8)/(1-x)^11.  - M. F. Hasler, Jan 19 2012
a(0)=0, a(1)=1, a(2)=34, a(3)=792, a(4)=10688, a(5)=90005, a(6)=533358, a(7)=2437848, a(8)=9156288, a(9)=29522961, a(10)=84293770, a(n)= 11*a(n-1)- 55*a(n-2)+165*a(n-3)-330*a(n-4)+462*a(n-5)-462*a(n-6)+ 330*a(n-7)- 165*a(n-8)+55*a(n-9)-11*a(n-10)+a(n-11). - Harvey P. Dale, Oct 20 2012
From Robert A. Russell, Oct 17 2020: (Start)
a(n) = A331350(n) - A331352(n) = (A331350(n) + A331353(n)) / 2 = A331352(n) + A331353(n).
a(n) = 1*C(n,1) + 32*C(n,2) + 693*C(n,3) + 7720*C(n,4) + 44150*C(n,5) + 138312*C(n,6) + 247380*C(n,7) + 252000*C(n,8) + 136080*C(n,9) + 30240*C(n,10), where the coefficient of C(n,k) is the number of unoriented colorings using exactly k colors. (End)

More terms from Vladeta Jovovic, Sep 02 2001

A331359 Number of unoriented colorings of the edges of a tesseract with n available colors.

Original entry on oeis.org

1, 11251322, 4825746875682, 48038446526132256, 60632984344185045000, 20725680132763499134746, 2876113738439693827763387, 206323339930086669420462592, 8941884949194537156253481511

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 unoriented colorings are the same if congruent; chiral pairs are counted as one. Also the number of unoriented colorings of the triangular faces of a regular 4-dimensional orthoplex {3,3,4} with n available colors.

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. A331358 (oriented), A331360 (chiral), A331361 (achiral).

Cf. A063843 (simplex), A331355 (orthoplex), A338953 (24-cell), A338965 (120-cell, 600-cell).

Table[(48n^4 + 64n^6 + 164n^8 + 32n^12 + 35n^16 + 24n^18 + 16n^20 + n^32)/384, {n, 1, 25}]

a(n) = (48*n^4 + 64*n^6 + 164*n^8 + 32*n^12 + 35*n^16 + 24*n^18 + 16*n^20 + n^32) / 384.
a(n) = C(n,1) + 11251320*C(n,2) + 4825713121719*C(n,3) + 48019143606137456*C(n,4) + 60392840368910627325*C(n,5) + 20362602706881512104770*C(n,6) + 2732305589004849709507320*C(n,7) + 183891356981584237730865120*C(n,8) + 7186781660980022442696996900*C(n,9) + 179941570950595830458653229400*C(n,10) + 3092495918800698593432175049200*C(n,11) + 38355721930679608007610435655200*C(n,12) + 356388702642082232961224416430400*C(n,13) + 2552262270629849366778056301033600*C(n,14) + 14398742619650679721666540905600000*C(n,15) + 65081946248235516086688061276416000*C(n,16) + 238774230958640327164289928460608000*C(n,17) + 718111905257279424242461614311808000*C(n,18) + 1783226074397879202567353905547520000*C(n,19) + 3674025240535453233878734112386560000*C(n,20) + 6297428247692138525542940292326400000*C(n,21) + 8984640042458034573900227275929600000*C(n,22) + 10651431202956156039912718487654400000*C(n,23) + 10448264801973961157855568414105600000*C(n,24) + 8418935641672774875938561280000000000*C(n,25) + 5510766716064148076659382317056000000*C(n,26) + 2882400456553496466714071801856000000*C(n,27) + 1175640370514915165746352603136000000*C(n,28) + 360177463966855890088916582400000000*C(n,29) + 77945658076061560043023564800000000*C(n,30) + 10621166594979816972895518720000000*C(n,31) + 685236554514826901477130240000000*C(n,32), where the coefficient of C(n,k) is the number of colorings using exactly k colors.
a(n) = A331358(n) - A331360(n) = (A331358(n) - A331361(n)) / 2 = A331360(n) + A331361(n).

A331354 Number of oriented colorings of the edges of a regular 4-dimensional orthoplex with n available colors.

Original entry on oeis.org

1, 90054, 1471640157, 1466049174160, 310441584462375, 24679078461920106, 997818989210621704, 24595659246351652992, 415450226822646218895, 5208333343963621522750, 51300691059764724112161, 414046079318115654521904

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 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.
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.
  Partition  Count  Even Cycle Indices
  4          6      8x_8^3
  31         8      4x_3^8 + 4x_6^4
  22         3      4x_1^4x_2^10 + 4x_4^6
  211        6      4x_1^2x_2^11 + 2x_1^4x_4^5 + 2x_2^2x_4^5
  1111       1      6x_1^4x_2^10 + x_1^24 + x_2^12

G. Royle, Partitions and Permutations
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. A331355 (unoriented), A331356 (chiral), A331357 (achiral).
Other polychora: A331350 (5-cell), A331358 (8-cell), A338952 (24-cell), A338964 (120-cell, 600-cell).
Row 4 of A337411 (orthoplex edges, orthotope ridges) and A337887 (orthotope faces, orthoplex peaks).

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}]

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.
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.
a(n) = A331355(n) + A331356(n) = 2*A331355(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).

A128767 Number of inequivalent n-colorings of the 4D hypercube under the full orthogonal group of the cube (of order 2^4*4! = 384).

Original entry on oeis.org

1, 402, 132102, 11756666, 405385550, 7416923886, 86986719477, 735192450952, 4834517667381, 26073250910950, 119759687845446, 481750080584202, 1733588303252702, 5673534527793146, 17109303241791825, 48047227408513056

Offset: 1

Ricardo Perez-Aguila (ricardo.perez.aguila(AT)gmail.com), Apr 04 2007

I assume this refers to colorings of the vertices of the cube. - N. J. A. Sloane, Apr 06 2007

Number of unoriented colorings of the 16 tetrahedral facets of a hyperoctahedron or of the 16 vertices of a tesseract (4-D cube) using up to n colors. Each chiral pair is counted as one when enumerating unoriented 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. - Robert A. Russell, Oct 03 2020

			a(2)=402 because there are 402 inequivalent 2-colorings of the 4D hypercube.

Banks, D. C.; Linton, S. A. & Stockmeyer, P. K. Counting Cases in Substitope Algorithms. IEEE Transactions on Visualization and Computer Graphics, Vol. 10, No. 4, pp. 371-384. 2004.
Perez-Aguila, Ricardo. Enumerating the Configurations in the n-Dimensional Orthogonal Polytopes Through Polya's Counting and A Concise Representation. Proceedings of the 3rd International Conference on Electrical and Electronics Engineering and XII Conference on Electrical Engineering ICEEE and CIE 2006, pp. 63-66.
Polya, G. & Read R. C. Combinatorial Enumeration of Groups, Graphs and Chemical Compounds. Springer-Verlag, 1987.

Banks, D. C.; Linton, S. A. & Stockmeyer, P. K., Counting Cases in Substitope Algorithms, IEEE Transactions on Visualization and Computer Graphics, Vol. 10, No. 4, pp. 371-384. 2004.
Perez-Aguila, Ricardo, Orthogonal Polytopes: Study and Application, PhD Thesis. Universidad de las Americas, Puebla. November, 2006.
Perez-Aguila, Ricardo, Enumerating the Configurations in the n-Dimensional Orthogonal Polytopes Through Polya's Counting and A Concise Representation, Proceedings of the 3rd International Conference on Electrical and Electronics Engineering and XII Conference on Electrical Engineering ICEEE and CIE 2006, pp. 63-66.
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. A337952 (oriented), A337954 (chiral), A337955 (achiral).
Other elements: A331359 (tesseract edges, hyperoctahedron faces), A331355 (tesseract faces, hyperoctahedron edges), A337957 (tesseract facets, hyperoctahedron vertices).
Other polychora: A000389(n+4) (4-simplex facets/vertices), A338949 (24-cell), A338965 (120-cell, 600-cell).
Row 4 of A325013 (orthoplex facets, orthotope vertices).

Table[(1/384)*( 48*n^2 + 180*n^4 + 48*n^6 + 83*n^8 + 12*n^10 + 12*n^12 + n^16),{n,30}]

a(n) = (1/384)*(48*n^2 + 180*n^4 + 48*n^6 + 83*n^8 + 12*n^10 + 12*n^12 + n^16)
G.f.: -x*(x +1)*(x^14 +384*x^13 +125020*x^12 +9439904*x^11 +213777216*x^10 +1821620108*x^9 +6527222787*x^8 +10098845160*x^7 +6527222787*x^6 +1821620108*x^5 +213777216*x^4 +9439904*x^3 +125020*x^2 +384*x +1) / (x -1)^17. [Colin Barker, Dec 04 2012]
From Robert A. Russell, Oct 03 2020: (Start)
a(n) = 1*C(n,1) + 400*C(n,2) + 130899*C(n,3) + 11230666*C(n,4) + 347919225*C(n,5) + 5158324560*C(n,6) + 43174480650*C(n,7) + 225086553300*C(n,8) + 775894225050*C(n,9) + 1831178115900*C(n,10) + 3008073915000*C(n,11) + 3439243962000*C(n,12) + 2685727044000*C(n,13) + 1366701336000*C(n,14) + 408648240000*C(n,15) + 54486432000*C(n,16), where the coefficient of C(n,k) is the number of unoriented colorings using exactly k colors.
a(n) = A337952(n) - A337954(n) = (A337952(n) + A337955(n)) / 2 = A337954(n) + A337955(n).
(End)

A337412 Array read by descending antidiagonals: T(n,k) is the number of unoriented 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, 21, 144, 1, 5, 55, 12111, 49127, 1, 6, 120, 358120, 740360358, 293122232, 1, 7, 231, 5131650, 733776248840, 3168520600399659, 25174334733080, 1, 8, 406, 45528756, 155261523065875, 314848558732420555904, 920040738175691418086226, 30035285091978202824, 1

Offset: 1

Robert A. Russell, Aug 26 2020

Each chiral pair is counted as one when enumerating unoriented 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 unoriented 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    21     55     120      231       406        666       1035 ...
1 144 12111 358120 5131650 45528756 288936634 1433251296 5887880415 ...
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. A337411 (oriented), A337413 (chiral), A337414 (achiral).
Rows 1-4 are A000027, A002817, A199406, A331355.
Cf. A327084 (simplex edges), A337408 (orthotope edges), A325005 (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; 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[]);
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)]
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) = A337411(n,k) - A337413(n,k) = (A337411(n,k) + A337414(n,k)) / 2 = A337413(n,k) + A337414(n,k).

A338953 Number of unoriented 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, 68774446639102959610154176, 5523164445430505961199114292414803649442426, 5448873034189827051954635848284422749083650351583379456, 10956401461402941741829572441752281718329313621842215239237500000

Offset: 1

Robert A. Russell, Nov 17 2020

Each chiral pair is counted as one when enumerating unoriented arrangements. The Schläfli symbol of the 24-cell is {3,4,3}. It has 24 octahedral facets. It is self-dual.

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

Cf. A338952 (oriented), A338954 (chiral), A338955 (achiral), A338957 (exactly n colors), A338949 (vertices, facets), A063843 (5-cell), A331359 (8-cell edges, 16-cell faces), A331355 (16-cell edges, 8-cell faces), A338965 (120-cell, 600-cell).

Table[(96n^8+144n^12+144n^16+128n^18+192n^20+228n^24+48n^32+32n^36+43n^48+72n^50+12n^52+12n^60+n^96)/1152,{n,15}]

a(n) = (96*n^8 + 144*n^12 + 144*n^16 + 128*n^18 + 192*n^20 + 228*n^24 +
48*n^32 + 32*n^36 + 43*n^48 + 72*n^50 + 12*n^52 + 12*n^60 + n^96) / 1152.
a(n) = Sum_{j=1..Min(n,96)} A338957(n) * binomial(n,j).
a(n) = A338952(n) - A338954(n) = (A338952(n) + A338955(n)) / 2 = A338954(n) + A338955(n).

A337957 Number of unoriented 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, 715, 3060, 10626, 31465, 82251, 194580, 424270, 864501, 1663740, 3049501, 5359095, 9078630, 14891626, 23738715, 36890001, 56031760, 83369265, 121747626, 174792640, 247073751, 344291325, 473490550, 643304376

Offset: 1

Robert A. Russell, Oct 03 2020

Each chiral pair is counted as one when enumerating unoriented 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. A337956 (oriented), A234249(n+1) (chiral), A337958 (achiral).
Other elements: A331355 (hyperoctahedron edges, tesseract faces), A331359 (hyperoctahedron faces, tesseract edges), A128767 (hyperoctahedron facets, tesseract vertices).
Other polychora: A000389(n+4) (5-cell), A338949 (24-cell), A338965 (120-cell, 600-cell).
Row 4 of A325005 (orthotope facets, orthoplex vertices).

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

a(n) = binomial(binomial(n+1,2)+3,4).
a(n) = n * (n+1) * (n^2 + n + 2) * (n^2 + n + 4) * (n^2 + n + 6) / 384.
a(n) = 1*C(n,1) + 13*C(n,2) + 84*C(n,3) + 297*C(n,4) + 600*C(n,5) + 690*C(n,6) + 420*C(n,7) + 105*C(n,8), where the coefficient of C(n,k) is the number of unoriented colorings using exactly k colors.
a(n) = A337956(n) - A234249(n+1) = (A337956(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 + 37*x^3 + 27*x^4 + 6*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)

A337888 Array read by descending antidiagonals: T(n,k) is the number of unoriented 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, 56, 49127, 1, 5, 220, 740360358, 314824532572147370464, 1, 6, 680, 733776248840, 38491882660671134164965704408524083, 38343035259947576596859948806931124970404417593861154473053467181056, 1

Offset: 2

Robert A. Russell, Sep 28 2020

Each chiral pair is counted as one when enumerating unoriented 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 unoriented 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        56          220             680              1771 ...
 1 49127 740360358 733776248840 155261523065875 12340612271439081 ...

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. A337887 (oriented), A337889 (chiral), A337890 (achiral).
Other elements: A325013 (vertices), A337408 (edges).
Other polytopes: A337884 (simplex), A337892 (orthoplex).
Rows 2-4 are A000027, A198833, A331355.

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; 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[]);
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)]
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) = A337887(n,k) - A337889(n,k) = (A337887(n,k) + A337890(n,k)) / 2 = A337889(n,k) + A337890(n,k).

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A063843 Number of n-multigraphs on 5 nodes.

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A331359 Number of unoriented colorings of the edges of a tesseract with n available colors.

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A331354 Number of oriented 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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A128767 Number of inequivalent n-colorings of the 4D hypercube under the full orthogonal group of the cube (of order 2^4*4! = 384).

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A337412 Array read by descending antidiagonals: T(n,k) is the number of unoriented colorings of the edges of a regular n-dimensional orthoplex (cross polytope) using k or fewer colors.

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A338953 Number of unoriented 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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