A325007 -id:A325007 - cp's OEIS frontend

A325006 Array read by descending antidiagonals: A(n,k) is the number of chiral pairs of colorings of the facets of a regular n-dimensional orthotope using up to k colors.

0, 1, 0, 3, 0, 0, 6, 3, 0, 0, 10, 15, 1, 0, 0, 15, 45, 20, 0, 0, 0, 21, 105, 120, 15, 0, 0, 0, 28, 210, 455, 210, 6, 0, 0, 0, 36, 378, 1330, 1365, 252, 1, 0, 0, 0, 45, 630, 3276, 5985, 3003, 210, 0, 0, 0, 0, 55, 990, 7140, 20475, 20349, 5005, 120, 0, 0, 0, 0, 66, 1485, 14190, 58905, 98280, 54264, 6435, 45, 0, 0, 0, 0

Offset: 1

Robert A. Russell, May 27 2019

Also called hypercube, n-dimensional cube, and measure polytope. For n=1, the figure is a line segment with two vertices. For n=2 the figure is a square with four edges. For n=3 the figure is a cube with six square faces. For n=4, the figure is a tesseract with eight cubic facets. The Schläfli symbol, {4,3,...,3}, of the regular n-dimensional orthotope (n>1) consists of a four followed by n-2 threes. Each of its 2n facets is an (n-1)-dimensional orthotope. The chiral colorings of its facets come in pairs, each the reflection of the other.

Also the number of chiral pairs of colorings of the vertices of a regular n-dimensional orthoplex using up to k colors.

			Array begins with A(1,1):
0 1 3  6  10   15     21       28        36         45          55 ...
0 0 3 15  45  105    210      378       630        990        1485 ...
0 0 1 20 120  455   1330     3276      7140      14190       26235 ...
0 0 0 15 210 1365   5985    20475     58905     148995      341055 ...
0 0 0  6 252 3003  20349    98280    376992    1221759     3478761 ...
0 0 0  1 210 5005  54264   376740   1947792    8145060    28989675 ...
0 0 0  0 120 6435 116280  1184040   8347680   45379620   202927725 ...
0 0 0  0  45 6435 203490  3108105  30260340  215553195  1217566350 ...
0 0 0  0  10 5005 293930  6906900  94143280  886163135  6358402050 ...
0 0 0  0   1 3003 352716 13123110 254186856 3190187286 29248649430 ...
For a(2,3)=3, each chiral pair consists of two adjacent edges of the square with one of the three colors.

Robert A. Russell, Table of n, a(n) for n = 1..325
Robin Chapman, answer to Coloring the faces of a hypercube, Math StackExchange, September 30, 2010.

Cf. A325004 (oriented), A325005 (unoriented), A325007 (achiral), A325010 (exactly k colors)
Other n-dimensional polytopes: A007318(k,n+1) (simplex), A325014 (orthoplex)
Rows 1-3 are A161680, A050534, A093566(n+1), A234249(n-1)

Table[Binomial[Binomial[d-n+1,2],n],{d,1,12},{n,1,d}] // Flatten

a(n, k) = binomial(binomial(k, 2), n)
array(rows, cols) = for(x=1, rows, for(y=1, cols, print1(a(x, y), ", ")); print(""))
/* Print initial 10 rows and 11 columns of array as follows: */
array(10, 11) \\ Felix Fröhlich, May 30 2019

A(n,k) = binomial(binomial(k,2),n).
A(n,k) = Sum_{j=1..2*n} A325010(n,j) * binomial(k,j).
A(n,k) = A325004(n,k) - A325005(n,k) = (A325004(n,k) - A325007(n,k)) / 2 = A325005(n,k) - A325007(n,k).
G.f. for row n: Sum{j=1..2*n} A325010(n,j) * x^j / (1-x)^(j+1).
Linear recurrence for row n: T(n,k) = Sum_{j=0..2*n} binomial(-2-j,2*n-j) * T(n,k-1-j).
G.f. for column k: (1+x)^binomial(k,2) - 1.

A325001 Array read by descending antidiagonals: A(n,k) is the number of achiral colorings of the facets (or vertices) of a regular n-dimensional simplex using up to k colors.

Original entry on oeis.org

1, 2, 1, 3, 4, 1, 4, 9, 5, 1, 5, 16, 15, 6, 1, 6, 25, 34, 21, 7, 1, 7, 36, 65, 56, 28, 8, 1, 8, 49, 111, 125, 84, 36, 9, 1, 9, 64, 175, 246, 210, 120, 45, 10, 1, 10, 81, 260, 441, 461, 330, 165, 55, 11, 1, 11, 100, 369, 736, 917, 792, 495, 220, 66, 12, 1

Offset: 1

Robert A. Russell, Mar 23 2019

For n=1, the figure is a line segment with two vertices. For n=2, the figure is a triangle with three edges. For n=3, the figure is a tetrahedron with four triangular faces. The Schläfli symbol, {3,...,3}, of the regular n-dimensional simplex consists of n-1 threes. Each of its n+1 facets is a regular (n-1)-dimensional simplex. An achiral coloring is the same as its reflection.

			The array begins with A(1,1):
  1  2  3   4   5    6    7     8     9    10    11     12     13 ...
  1  4  9  16  25   36   49    64    81   100   121    144    169 ...
  1  5 15  34  65  111  175   260   369   505   671    870   1105 ...
  1  6 21  56 125  246  441   736  1161  1750  2541   3576   4901 ...
  1  7 28  84 210  461  917  1688  2919  4795  7546  11452  16848 ...
  1  8 36 120 330  792 1715  3424  6399 11320 19118  31032  48672 ...
  1  9 45 165 495 1287 3003  6434 12861 24265 43593  75087 124683 ...
  1 10 55 220 715 2002 5005 11440 24309 48610 92323 167740 293215 ...
  ...
For A(2,2)=4, the triangle may have 0, 1, 2, or 3 edges of one color.

Robert A. Russell, Table of n, a(n) for n = 1..1275

Cf. A324999 (oriented), A325000 (unoriented), A325000(n,k-n) (chiral), A325003 (exactly k colors), A327086 (edges, ridges), A337886 (faces, peaks), A325007 (orthotope facets, orthoplex vertices), A325015 (orthoplex facets, orthotope vertices).
Rows 1-4 are A000027, A000290, A006003, A132366(n-1).
Column 2 is A162880.

Table[Binomial[d+1,n+1] - Binomial[d+1-n,n+1], {d,1,15}, {n,1,d}] // Flatten

A(n,k) = binomial(n+k,n+1) - binomial(k,n+1).
A(n,k) = Sum_{j=1..n} A325003(n,j) * binomial(k,j).
A(n,k) = 2*A325000(n,k) - A324999(n,k) = A324999(n,k) - 2*A325000(n,k-n) = A325000(n,k) - A325000(n,k-n).
G.f. for row n: (x - x^(n+1)) / (1-x)^(n+2).
Linear recurrence for row n: A(n,k) = Sum_{j=1..n+1} -binomial(j-n-2,j) * A(n,k-j).
G.f. for column k: (1 - (1-x^2)^k) / (x*(1-x)^k).

A325004 Array read by descending antidiagonals: A(n,k) is the number of oriented colorings of the facets of a regular n-dimensional orthotope using up to k colors.

Original entry on oeis.org

1, 4, 1, 9, 6, 1, 16, 24, 10, 1, 25, 70, 57, 15, 1, 36, 165, 240, 126, 21, 1, 49, 336, 800, 730, 252, 28, 1, 64, 616, 2226, 3270, 2008, 462, 36, 1, 81, 1044, 5390, 11991, 11880, 5006, 792, 45, 1, 100, 1665, 11712, 37450, 56133, 38970, 11440, 1287, 55, 1

Offset: 1

Robert A. Russell, Mar 23 2019

Also called hypercube, n-dimensional cube, and measure polytope. For n=1, the figure is a line segment with two vertices. For n=2 the figure is a square with four edges. For n=3 the figure is a cube with six square faces. For n=4, the figure is a tesseract with eight cubic facets. The Schläfli symbol, {4,3,...,3}, of the regular n-dimensional orthotope (n>1) consists of a four followed by n-2 threes. Each of its 2n facets is an (n-1)-dimensional orthotope. 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 vertices of a regular n-dimensional orthoplex using up to k colors.

			Array begins with A(1,1):
1  4    9    16     25      36       49        64        81        100 ...
1  6   24    70    165     336      616      1044      1665       2530 ...
1 10   57   240    800    2226     5390     11712     23355      43450 ...
1 15  126   730   3270   11991    37450    102726    253485     573265 ...
1 21  252  2008  11880   56133   221725    756288   2283876    6228145 ...
1 28  462  5006  38970  235235  1161832   4873128  17838492   58208920 ...
1 36  792 11440 116400  894465  5495896  28162368 124122780  481650400 ...
1 45 1287 24310 319815 3114540 23739310 148116618 782798490 3596651740 ...
For A(1,2) = 4, the two achiral colorings use just one of the two colors for both vertices; the chiral pair uses two colors.

Robert A. Russell, Table of n, a(n) for n = 1..325
Robin Chapman, answer to Coloring the faces of a hypercube, Math StackExchange, September 30, 2010.

Cf. A325005 (unoriented), A325006 (chiral), A325007 (achiral), A325008 (exactly k colors)
Other n-dimensional polytopes: A324999 (simplex), A325012 (orthoplex)
Rows 1-3 are A000290, A006528, A047780.

Table[Binomial[Binomial[d-n+2,2]+n-1,n]+Binomial[Binomial[d-n+1,2],n],{d,1,11},{n,1,d}] // Flatten

A(n,k) = binomial(binomial(k+1,2) + n-1, n) + binomial(binomial(k,2),n).
A(n,k) = Sum_{j=1..2n} A325008(n,j) * binomial(k,j).
A(n,k) = A325005(n,k) + A325006(n,k) = 2*A325005(n,k) - A325007(n,k) = 2*A325006(n,k) + A325007(n,k).
G.f. for row n: Sum{j=1..2n} A325008(n,j) * x^j / (1-x)^(j+1).
Linear recurrence for row n: T(n,k) = Sum_{j=0..2n} binomial(-2-j,2n-j) * T(n,k-1-j).
G.f. for column k: 1/(1-x)^binomial(k+1,2) + (1+x)^binomial(k,2) - 2.

A325005 Array read by descending antidiagonals: A(n,k) is the number of unoriented colorings of the facets of a regular n-dimensional orthotope using up to k colors.

Original entry on oeis.org

1, 3, 1, 6, 6, 1, 10, 21, 10, 1, 15, 55, 56, 15, 1, 21, 120, 220, 126, 21, 1, 28, 231, 680, 715, 252, 28, 1, 36, 406, 1771, 3060, 2002, 462, 36, 1, 45, 666, 4060, 10626, 11628, 5005, 792, 45, 1, 55, 1035, 8436, 31465, 53130, 38760, 11440, 1287, 55, 1

Offset: 1

Robert A. Russell, Mar 23 2019

Also called hypercube, n-dimensional cube, and measure polytope. For n=1, the figure is a line segment with two vertices. For n=2 the figure is a square with four edges. For n=3 the figure is a cube with six square faces. For n=4, the figure is a tesseract with eight cubic facets. The Schläfli symbol, {4,3,...,3}, of the regular n-dimensional orthotope (n>1) consists of a four followed by n-2 threes. Each of its 2n facets is an (n-1)-dimensional orthotope. Two unoriented colorings are the same if congruent; chiral pairs are counted as one.

Also the number of unoriented colorings of the vertices of a regular n-dimensional orthoplex using up to k colors.

			Array begins with A(1,1):
1  3    6    10     15      21       28        36        45         55 ...
1  6   21    55    120     231      406       666      1035       1540 ...
1 10   56   220    680    1771     4060      8436     16215      29260 ...
1 15  126   715   3060   10626    31465     82251    194580     424270 ...
1 21  252  2002  11628   53130   201376    658008   1906884    5006386 ...
1 28  462  5005  38760  230230  1107568   4496388  15890700   50063860 ...
1 36  792 11440 116280  888030  5379616  26978328 115775100  436270780 ...
1 45 1287 24310 319770 3108105 23535820 145008513 752538150 3381098545 ...
For A(1,2) = 3, the two achiral colorings use just one of the two colors for both vertices; the chiral pair uses one color for each vertex.

Robert A. Russell, Table of n, a(n) for n = 1..325
Robin Chapman, answer to Coloring the faces of a hypercube, Math StackExchange, September 30, 2010.

Cf. A325004 (oriented), A325006 (chiral), A325007 (achiral), A325009 (exactly k colors).
Other n-dimensional polytopes: A325000 (simplex), A325013 (orthoplex).
Rows 1-3 are A000217, A002817, A198833.

Table[Binomial[Binomial[d-n+2,2]+n-1,n],{d,1,11},{n,1,d}] // Flatten

A(n,k) = binomial(n + binomial(k+1,2) - 1, n).
A(n,k) = Sum_{j=1..2n} A325009(n,j) * binomial(k,j).
A(n,k) = A325004(n,k) - A325006(n,k) = (A325004(n,k) + A325007(n,k)) / 2 = A325006(n,k) + A325007(n,k).
G.f. for row n: Sum_{j=1..2n} A325009(n,j) * x^j / (1-x)^(j+1).
Linear recurrence for row n: T(n,k) = Sum_{j=0..2n} binomial(-2-j,2n-j) * T(n,k-1-j).
G.f. for column k: 1/(1-x)^binomial(k+1,2) - 1.

A325015 Array read by descending antidiagonals: T(n,k) is the number of achiral colorings of the facets of a regular n-dimensional orthoplex using up to k colors.

Original entry on oeis.org

1, 2, 1, 3, 6, 1, 4, 18, 21, 1, 5, 40, 201, 308, 1, 6, 75, 1076, 34128, 180342, 1, 7, 126, 4025, 1056576, 2945136213, 366975285216, 1, 8, 196, 11901, 15303750, 2932338749408, 103863386269870076808, 10316179427644325573474464, 1

Offset: 1

Robert A. Russell, May 27 2019

Also called cross polytope and hyperoctahedron. For n=1, the figure is a line segment with two vertices. For n=2 the figure is a square with four edges. For n=3 the figure is an octahedron with eight triangular faces. For n=4, the figure is a 16-cell with sixteen tetrahedral facets. The Schläfli symbol, {3,...,3,4}, of the regular n-dimensional orthoplex (n>1) consists of n-2 threes followed by a four. Each of its 2^n facets is an (n-1)-dimensional simplex. An achiral coloring is identical to its reflection.

Also the number of achiral colorings of the vertices of a regular n-dimensional orthotope (cube) using up to k colors.

			Array begins with T(1,1):
1   2     3       4        5         6         7          8 ...
1   6    18      40       75       126       196        288 ...
1  21   201    1076     4025     11901     29841      66256 ...
1 308 34128 1056576 15303750 136236276 865711763 4296782848 ...
...
For T(2,2)=6, two squares have all edges the same color, two have three edges the same color, one has opposite edges the same color, and one has opposite edges different colors.

Robert A. Russell, Table of n, a(n) for n = 1..78
E. M. Palmer and R. W. Robinson, Enumeration under two representations of the wreath product, Acta Math., 131 (1973), 123-143.
Wikipedia, Cross-polytope

Cf. A325012 (oriented), A325013 (unoriented), A325014 (chiral), A325019 (exactly k colors).
Other n-dimensional polytopes: A325001 (simplex), A325007 (orthotope).
Rows 1-2 are A000027, A002411.
Cf. A000048, A001037.

a48[n_] := a48[n] = DivisorSum[NestWhile[#/2&, n, EvenQ], MoebiusMu[#]2^(n/#)&]/(2n); (* A000048 *)
a37[n_] := a37[n] = DivisorSum[n, MoebiusMu[n/#]2^#&]/n; (* A001037 *)
CI0[{n_Integer}] := CI0[{n}] = CI[Transpose[If[EvenQ[n], p2 = IntegerExponent[n, 2]; sub = Divisors[n/2^p2]; {2^(p2+1) sub, a48 /@ (2^p2 sub) }, sub = Divisors[n]; {sub, a37 /@ sub}]]] 2^(n-1); (* even perm. *)
CI1[{n_Integer}] := CI1[{n}] = CI[sub = Divisors[n]; Transpose[If[EvenQ[n], {sub, a37 /@ sub}, {2 sub, a48 /@ sub}]]] 2^(n-1); (* odd perm. *)
compress[x : {{, } ...}] := (s = Sort[x]; For[i = Length[s], i > 1, i -= 1, If[s[[i, 1]]==s[[i-1, 1]], s[[i-1, 2]] += s[[i, 2]]; s = Delete[s, i], Null]]; s)
cix[{a_, b_}, {c_, d_}] := {LCM[a, c], (a b c d)/LCM[a, c]};
Unprotect[Times]; Times[CI[a_List], CI[b_List]] :=  (* combine *) CI[compress[Flatten[Outer[cix, a, b, 1], 1]]]; Protect[Times];
CI0[p_List] := CI0[p] = Expand[CI0[Drop[p, -1]] CI0[{Last[p]}] + CI1[Drop[p, -1]] CI1[{Last[p]}]]
CI1[p_List] := CI1[p] = Expand[CI0[Drop[p, -1]] CI1[{Last[p]}] + CI1[Drop[p, -1]] CI0[{Last[p]}]]
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[(CI1[#] pc[#]) & /@ IntegerPartitions[n]])/(n! 2^(n - 1))] /. CI[l_List] :> j^(Total[l][[2]])
array[n_, k_] := row[n] /. j -> k
Table[array[n, d-n+1], {d, 1, 10}, {n, 1, d}] // Flatten

The algorithm used in the Mathematica program below assigns each permutation of the axes to a partition of n. It then determines the number of permutations for each partition and the cycle index for each partition.
T(n,k) = 2*A325013(n,k) - A325012(n,k) = A325012(n,k) - 2*A325014(n,k) = A325013(n,k) - A325014(n,k).
T(n,k) = Sum_{j=1..3*2^(n-2)} A325019(n,j) * binomial(k,j).

A325011 Triangle read by rows: T(n,k) is the number of achiral colorings of the facets of a regular n-dimensional orthotope using exactly k colors. Row n has 2n columns.

Original entry on oeis.org

1, 0, 1, 4, 3, 0, 1, 8, 28, 36, 15, 0, 1, 13, 84, 282, 465, 360, 105, 0, 1, 19, 192, 1110, 3711, 7080, 7560, 4200, 945, 0, 1, 26, 381, 3320, 17875, 60159, 126728, 165900, 130725, 56700, 10395, 0, 1, 34, 687, 8484, 66525, 340929, 1158102, 2624748, 3964905, 3931200, 2453220, 873180, 135135, 0

Offset: 1

Robert A. Russell, May 27 2019

Also called hypercube, n-dimensional cube, and measure polytope. For n=1, the figure is a line segment with two vertices. For n=2 the figure is a square with four edges. For n=3 the figure is a cube with six square faces. For n=4, the figure is a tesseract with eight cubic facets. The Schläfli symbol, {4,3,...,3}, of the regular n-dimensional orthotope (n>1) consists of a four followed by n-2 threes. Each of its 2n facets is an (n-1)-dimensional orthotope. An achiral coloring is identical to its reflection.

Also the number of achiral colorings of the vertices of a regular n-dimensional orthoplex using exactly k colors.

			Table begins with T(1,1):
 1  0
 1  4   3    0
 1  8  28   36    15     0
 1 13  84  282   465   360    105      0
 1 19 192 1110  3711  7080   7560   4200    945     0
 1 26 381 3320 17875 60159 126728 165900 130725 56700 10395 0
For T(2,3)=3, each of the three chiral pairs has two opposite edges with the same color.

Robert A. Russell, Table of n, a(n) for n = 1..132

Cf. A325008 (oriented), A325009 (unoriented), A325010 (chiral), A325007 (up to k colors).

Other n-dimensional polytopes: A325003 (simplex), A325019 (orthoplex).

Table[Sum[Binomial[-j-2,k-j-1] Binomial[n + Binomial[j+2,2]-1, n], {j,0,k-1}] - Sum[Binomial[j-k-1,j] Binomial[Binomial[k-j,2],n],{j,0,k-2}], {n,1,10},{k,1,2n}] // Flatten

T(n,k) = Sum_{j=0..k-1} binomial(-j-2,k-j-1) * binomial(n + binomial(j+2,2)-1, n) - Sum_{j=0..k-2} binomial(j-k-1,j) * binomial(binomial(k-j,2),n).

T(n,k) = 2*A325009(n,k) - A325008(n,k) = A325008(n,k) - 2*A325010(n,k) = A325009(n,k) - A325010(n,k).

A337414 Array read by descending antidiagonals: T(n,k) is the number of achiral 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, 18, 70, 1, 5, 40, 1407, 8200, 1, 6, 75, 12480, 9080559, 12804908, 1, 7, 126, 69050, 1503323520, 4906480368591, 304899216832, 1, 8, 196, 281946, 81461669375, 48226825456539776, 187380251418565888983, 103685962258536432, 1

Offset: 1

Robert A. Russell, Aug 26 2020

An achiral arrangement is identical to its reflection. 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 achiral 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       10 ...
1  6   18    40    75    126    196     288     405      550 ...
1 70 1407 12480 69050 281946 931490 2632512 6598935 15041950 ...
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), A337412 (unoriented), A337413 (chiral).
Rows 1-4 are A000027, A002411, A331351, A331357.
Cf. A327086 (simplex edges), A337410 (orthotope edges), A325007 (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}]],0,(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-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) = 2*A337412(n,k) - A337411(n,k) = A337411(n,k) - 2*A337413(n,k) = A337412(n,k) - A337413(n,k).

A337898 Number of achiral colorings of the 6 square faces of a cube or the 6 vertices of a regular octahedron using n or fewer colors.

Original entry on oeis.org

1, 10, 55, 200, 560, 1316, 2730, 5160, 9075, 15070, 23881, 36400, 53690, 77000, 107780, 147696, 198645, 262770, 342475, 440440, 559636, 703340, 875150, 1079000, 1319175, 1600326, 1927485, 2306080, 2741950, 3241360

Offset: 1

Robert A. Russell, Sep 28 2020

An achiral coloring is identical to its reflection. The Schläfli symbols for the cube and regular octahedron are {4,3} and {3,4} respectively. They are mutually dual.
There are 24 elements in the automorphism group of the regular octahedron/cube that are not in the rotation group. They divide into five conjugacy classes. The first formula is obtained by averaging the cube face (octahedron vertex) cycle indices after replacing x_i^j with n^j according to the Pólya enumeration theorem.
  Conjugacy Class     Count    Odd Cycle Indices
  Inversion              1     x_2^3
  Vertex rotation*       8     x_6^1           Asterisk indicates that the
  Edge rotation*         6     x_1^2x_2^2      operation is followed by an
  Small face rotation*   6     x_2^1x_4^1      inversion.
  Large face rotation*   3     x_1^4x_2^1

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A047780 (oriented), A198833 (unoriented), A093566(n+1) (chiral).
Other elements: A331351 (edges), A337897 (cube vertices/octahedron faces).
Other polyhedra: A006003 (simplex), A337962 (dodecahedron faces, icosahedron vertices), A337960 (icosahedron faces, dodecahedron vertices).
Row 3 of A325007 (orthotope facets, orthoplex vertices) and A337890 (orthotope faces, orthoplex peaks).

Table[n(1+n)(2+n)(4-3n+3n^2)/24, {n, 35}]
LinearRecurrence[{6,-15,20,-15,6,-1},{1,10,55,200,560,1316},40] (* Harvey P. Dale, Feb 15 2022 *)

a(n)=n*(n+1)*(n+2)*(3*n^2-3*n+4)/24 \\ Charles R Greathouse IV, Oct 21 2022

a(n) = n * (n+1) * (n+2) * (3*n^2 - 3*n + 4) / 24.
a(n) = 1*C(n,1) + 8*C(n,2) + 28*C(n,3) + 36*C(n,4) + 15*C(n,5), where the coefficient of C(n,k) is the number of achiral colorings using exactly k colors.
a(n) = 2*A198833(n) - A047780(n) = A047780(n) - 2*A093566(n+1) = A198833(n) - A093566(n+1).
G.f.: x * (x + 4*x^2 + 10*x^3) / (1-x)^6.
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6). - Wesley Ivan Hurt, Sep 30 2020

A337958 Number of achiral 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, 700, 2850, 9261, 25480, 61776, 135675, 275275, 523446, 943020, 1623076, 2686425, 4298400, 6677056, 10104885, 14942151, 21641950, 30767100, 43008966, 59208325, 80378376, 107730000, 142699375, 186978051, 242545590

Offset: 1

Robert A. Russell, Oct 03 2020

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.

Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A337956 (oriented), A337956 (unoriented), A234249(n+1) (chiral).
Other elements: A331357 (hyperoctahedron edges, tesseract faces), A331361 (hyperoctahedron faces, tesseract edges), A337955 (hyperoctahedron facets, tesseract vertices).
Other polychora: A132366(n-1) (5-cell), A338951 (24-cell), A338967 (120-cell, 600-cell).
Row 4 of A325007 (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^2 * (n+1)^2 * (n+3) * (n^2 -2n +4) / 48.
a(n) = 1*C(n,1) + 13*C(n,2) + 84*C(n,3) + 282*C(n,4) + 465*C(n,5) + 360*C(n,6) + 105*C(n,7), where the coefficient of C(n,k) is the number of achiral colorings using exactly k colors.
a(n) = 2*A337957(n) - A337956(n) = A337956(n) - 2 * A234249(n+1) = A337957(n) - A234249(n+1).
From Stefano Spezia, Oct 04 2020: (Start)
G.f.: x*(1 + 7*x + 34*x^2 + 56*x^3 + 8*x^4 - x^5)/(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)

A337894 Array read by descending antidiagonals: T(n,k) is the number of achiral colorings of the faces of a regular n-dimensional orthoplex (cross polytope) using k or fewer colors.

Original entry on oeis.org

1, 2, 1, 3, 21, 1, 4, 201, 93024, 1, 5, 1076, 294157089, 199556208371776, 1, 6, 4025, 91983927296, 1370366433970979158839987, 346179533768149850758531729588224, 1

Offset: 2

Robert A. Russell, Sep 28 2020

An achiral arrangement is identical to its reflection. For n=2, the figure is a square with one square face. For n=3, the figure is an octahedron with 8 triangular faces. For higher n, the number of triangular faces is 8*C(n,3).

Also the number of achiral colorings of the peaks of an n-dimensional orthotope (hypercube). A peak is an (n-3)-dimensional orthotope.

			Table begins with T(2,1):
1     2         3           4             5               6 ...
1    21       201        1076          4025           11901 ...
1 93024 294157089 91983927296 7960001890625 304914963625056 ...

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. A337891 (oriented), A337892 (unoriented), A337893 (chiral).
Other elements: A325007 (vertices), A337414 (edges).
Other polytopes: A337886 (simplex), A337890 (orthotope).
Rows 2-4 are A000027, A337897, A331361.

m=2; (* dimension of color element, here a 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, 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}]],0,(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-1))]
array[n_, k_] := row[n] /. b -> k
Table[array[n,d+m-n], {d,7}, {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) = 2*A337892(n,k) - A337891(n,k) = A337891(n,k) - 2*A337893(n,k) = A337892(n,k) - A337893(n,k).

cp's OEIS Frontend

A325006 Array read by descending antidiagonals: A(n,k) is the number of chiral pairs of colorings of the facets of a regular n-dimensional orthotope using up to k colors.

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A325001 Array read by descending antidiagonals: A(n,k) is the number of achiral colorings of the facets (or vertices) of a regular n-dimensional simplex using up to k colors.

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A325004 Array read by descending antidiagonals: A(n,k) is the number of oriented colorings of the facets of a regular n-dimensional orthotope using up to k colors.

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A325005 Array read by descending antidiagonals: A(n,k) is the number of unoriented colorings of the facets of a regular n-dimensional orthotope using up to k colors.

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A325015 Array read by descending antidiagonals: T(n,k) is the number of achiral colorings of the facets of a regular n-dimensional orthoplex using up to k colors.

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A325011 Triangle read by rows: T(n,k) is the number of achiral colorings of the facets of a regular n-dimensional orthotope using exactly k colors. Row n has 2n columns.

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A337414 Array read by descending antidiagonals: T(n,k) is the number of achiral colorings of the edges of a regular n-dimensional orthoplex (cross polytope) using k or fewer colors.

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A337898 Number of achiral colorings of the 6 square faces of a cube or the 6 vertices of a regular octahedron using n or fewer colors.