A325008 -id:A325008 - cp's OEIS frontend

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.

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.

A325009 Triangle read by rows: T(n,k) is the number of unoriented 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, 1, 1, 4, 6, 3, 1, 8, 29, 52, 45, 15, 1, 13, 84, 297, 600, 690, 420, 105, 1, 19, 192, 1116, 3933, 8661, 11970, 10080, 4725, 945, 1, 26, 381, 3321, 18080, 63919, 150332, 236978, 247275, 163800, 62370, 10395, 1, 34, 687, 8484, 66645, 346644, 1231857, 3052008, 5316885, 6483330, 5415795, 2952180, 945945, 135135

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. 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 exactly k colors.

			The triangle begins with T(1,1):
1  1
1  4   6    3
1  8  29   52    45    15
1 13  84  297   600   690    420    105
1 19 192 1116  3933  8661  11970  10080   4725    945
1 26 381 3321 18080 63919 150332 236978 247275 163800 62370 10395
For T(2,2)=4, there are two squares with just one edge for one color, one square with opposite edges the same color, and one square with opposite edges different colors.

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

Cf. A325008 (oriented), A325010 (chiral), A325011 (achiral), A325005 (up to k colors).

Other n-dimensional polytopes: A007318(n,k-1) (simplex), A325017 (orthoplex).

Table[Sum[Binomial[-j-2,k-j-1]Binomial[n+Binomial[j+2,2]-1,n],{j,0,k-1}],{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).

T(n,k) = A325009(n,k) + A325010(n,k) = 2*A325009(n,k) - A325011(n,k) = 2*A325010(n,k) + A325011(n,k).

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).

A325016 Triangle read by rows: T(n,k) is the number of oriented colorings of the facets of a regular n-dimensional orthoplex using exactly k colors. Row n has 2^n columns.

Original entry on oeis.org

1, 2, 1, 4, 9, 6, 1, 21, 267, 1718, 5250, 7980, 5880, 1680, 1, 494, 228591, 21539424, 685479375, 10257064650, 86151316860, 449772354360, 1551283253100, 3661969537800, 6015983173200, 6878457986400, 5371454088000, 2733402672000, 817296480000, 108972864000

Offset: 1

Robert A. Russell, May 28 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. 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 orthotope (cube) using exactly k colors.

			Triangle begins with T(1,1):
  1  2
  1  4   9    6
  1 21 267 1718 5250 7980 5880 1680
For T(2,2)=4, two squares 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..510, rows 1..8, flattened.
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. A325017 (unoriented), A325018 (chiral), A325019 (achiral), A325012 (up to k colors).
Other n-dimensional polytopes: A325002 (simplex), A325008 (orthotope).
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, (a37 /@ sub)/2}]]] 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[(CI0[#] pc[#]) & /@ IntegerPartitions[n]])/(n! 2^(n - 1))] /. CI[l_List] :> j^(Total[l][[2]])
array[n_, k_] := row[n] /. j -> k (* A325012 *)
Table[LinearSolve[Table[Binomial[i,j],{i,1,2^n},{j,1,2^n}],Table[array[n,k],{k,1,2^n}]],{n,1,6}] // Flatten

A325012(n,k) = Sum_{j=1..2^n} T(n,j) * binomial(k,j).

T(n,k) = A325017(n,k) + A325018(n,k) = 2*A325017(n,k) - A325019(n,k) = 2*A325018(n,k) + A325019(n,k).

A325002 Triangle read by rows: T(n,k) is the number of oriented colorings of the facets (or vertices) of a regular n-dimensional simplex using exactly k colors.

Original entry on oeis.org

1, 2, 1, 2, 2, 1, 3, 3, 2, 1, 4, 6, 4, 2, 1, 5, 10, 10, 5, 2, 1, 6, 15, 20, 15, 6, 2, 1, 7, 21, 35, 35, 21, 7, 2, 1, 8, 28, 56, 70, 56, 28, 8, 2, 1, 9, 36, 84, 126, 126, 84, 36, 9, 2, 1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 2, 1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 2

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 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. Two oriented colorings are the same if one is a rotation of the other; chiral pairs are counted as two. The only chiral pair occurs when k=n+1; for k <= n all the colorings are achiral.

			Triangle begins with T(1,1):
1  2
1  2   2
1  3   3   2
1  4   6   4    2
1  5  10  10    5    2
1  6  15  20   15    6    2
1  7  21  35   35   21    7    2
1  8  28  56   70   56   28    8    2
1  9  36  84  126  126   84   36    9    2
1 10  45 120  210  252  210  120   45   10    2
1 11  55 165  330  462  462  330  165   55   11    2
1 12  66 220  495  792  924  792  495  220   66   12   2
1 13  78 286  715 1287 1716 1716 1287  715  286   78  13   2
1 14  91 364 1001 2002 3003 3432 3003 2002 1001  364  91  14  2
1 15 105 455 1365 3003 5005 6435 6435 5005 3003 1365 455 105 15 2
For T(3,2)=3, the tetrahedron may have one, two, or three faces of one color.

Robert A. Russell, Table of n, a(n) for n = 1..1325

Cf. A007318(n,k-1) (unoriented), A325003 (achiral), A325001 (up to k colors).

Other n-dimensional polytopes: A325008 (orthotope), A325016 (orthoplex).

Table[Binomial[n,k-1] + Boole[k==n+1], {n,1,15}, {k,1,n+1}] \\ Flatten

T(n,k) = binomial(n,k-1) + [k==n+1] = A007318(n,k-1) + [k==n+1].
T(n,k) = 2*A007318(n,k-1) - A325003(n,k) = [k==n+1] + A325003(n,k).
G.f. for row n: x * (1+x)^n + x^(n+1).
G.f. for column k>1: x^(k-1)/(1-x)^k + x^(k-1).

A325010 Triangle read by rows: T(n,k) is the number of chiral pairs of colorings of the facets of a regular n-dimensional orthotope using exactly k colors. Row n has 2n columns.

Original entry on oeis.org

0, 1, 0, 0, 3, 3, 0, 0, 1, 16, 30, 15, 0, 0, 0, 15, 135, 330, 315, 105, 0, 0, 0, 6, 222, 1581, 4410, 5880, 3780, 945, 0, 0, 0, 1, 205, 3760, 23604, 71078, 116550, 107100, 51975, 10395, 0, 0, 0, 0, 120, 5715, 73755, 427260, 1351980, 2552130, 2962575, 2079000, 810810, 135135

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 exactly k colors.

			The triangle begins with T(1,1):
 0 1
 0 0 3  3
 0 0 1 16  30   15
 0 0 0 15 135  330   315    105
 0 0 0  6 222 1581  4410   5880    3780     945
 0 0 0  1 205 3760 23604  71078  116550  107100   51975   10395
 0 0 0  0 120 5715 73755 427260 1351980 2552130 2962575 2079000 810810 135135
For T(2,3)=3, the three squares have the two edges with the same color adjacent.

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

Cf. A325008 (oriented), A325009 (unoriented), A325011 (achiral), A325006 (up to k colors).

Other n-dimensional polytopes: A325018 (orthoplex).

Table[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-2} binomial(j-k-1,j) * binomial(binomial(k-j,2),n).

T(n,k) = A325008(n,k) - A325009(n,k) = (A325008(n,k) - A325011(n,k)) / 2 = A325009(n,k) - A325011(n,k).

A338142 Triangle read by rows: T(n,k) is the number of oriented colorings of the edges of a regular n-D orthotope (or ridges of a regular n-D orthoplex) using exactly k colors. Row n has n*2^(n-1) columns.

Original entry on oeis.org

1, 1, 4, 9, 6, 1, 216, 22164, 613804, 6901425, 39713430, 131754420, 267165360, 336798000, 257796000, 109771200, 19958400, 1, 22409618, 9651132365418, 96038196404417832, 120785673234798359850

Offset: 1

Robert A. Russell, Oct 12 2020

Each chiral pair is counted as two when enumerating oriented arrangements. A ridge is an (n-2)-face of an n-D polytope. For n=1, the figure is a line segment with one edge. For n=2, the figure is a square with 4 edges (vertices). For n=3, the figure is a cube (octahedron) with 12 edges. The number of edges (ridges) is n*2^(n-1). The Schläfli symbols for the n-D orthotope (hypercube) and the n-D orthoplex (hyperoctahedron, cross polytope) are {4,...,3,3} and {3,3,...,4} respectively, with n-2 3's in each case. The figures are mutually dual.

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).

			Triangle begins with T(1,1):
  1
  1   4     9      6
  1 216 22164 613804 6901425 39713430 131754420 267165360 336798000
  ...

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. A338143 (unoriented), A338144 (chiral), A338145 (achiral), A337407 (k or fewer colors), A325016 (orthotope vertices, orthoplex facets).

Cf. A327087 (simplex), A338146 (orthoplex edges, orthotope ridges).

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, 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[LinearSolve[Table[Binomial[i,j],{i,2^(n-m)Binomial[n,m]},{j,2^(n-m)Binomial[n,m]}], Table[array[n,k],{k,2^(n-m)Binomial[n,m]}]], {n,m,m+4}] // Flatten

A337407(n,k) = Sum_{j=1..n*2^(n-1)} T(n,j) * binomial(k,j).
T(n,k) = A338143(n,k) + A338144(n,k) = 2*A338143(n,k) - A338145(n,k) = 2*A338144(n,k) + A338145(n,k).
T(2,k) = A338146(2,k) = A325016(2,k) = A325008(2,k); T(3,k) = A338146(3,k).

A338146 Triangle read by rows: T(n,k) is the number of oriented colorings of the edges of a regular n-D orthoplex (or ridges of a regular n-D orthotope) using exactly k colors. Row 1 has 1 column; row n>1 has 2n(n-1) columns.

Original entry on oeis.org

1, 1, 4, 9, 6, 1, 216, 22164, 613804, 6901425, 39713430, 131754420, 267165360, 336798000, 257796000, 109771200, 19958400, 1, 90052, 1471369998, 1460163153852, 303126054092610, 22838390261305920, 831533453035309605

Offset: 1

Robert A. Russell, Oct 12 2020

Each chiral pair is counted as two when enumerating oriented arrangements. A ridge is an (n-2)-face of an n-D polytope. For n=1, the figure is a line segment with one edge. For n=2, the figure is a square with 4 edges (vertices). For n=3, the figure is an octahedron (cube) with 12 edges. For n>1, the number of edges (ridges) is 2*n*(n-1). The Schläfli symbols for the n-D orthotope (hypercube) and the n-D orthoplex (hyperoctahedron, cross polytope) are {4,3,...,3,3} and {3,3,...,3,4} respectively, with n-2 3's in each case. The figures are mutually dual.

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).

			Triangle begins with T(1,1):
  1
  1   4     9      6
  1 216 22164 613804 6901425 39713430 131754420 267165360 336798000
  ...
For T(2,3)=9, the 3 achiral colorings are ABAC, ABCB, and ACBC. The three chiral pairs are AABC-AACB, ABBC-ACBB, and ABCC-ACCB.

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. A338147 (unoriented), A338148 (chiral), A338149 (achiral), A337411 (k or fewer colors), A325008 (orthoplex vertices, orthotope facets).

Cf. A327087 (simplex), A338142 (orthotope edges, orthoplex ridges).

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
Join[{{1}},Table[LinearSolve[Table[Binomial[i,j],{i,2^(m+1)Binomial[n,m+1]},{j,2^(m+1)Binomial[n,m+1]}], Table[array[n,k],{k,2^(m+1)Binomial[n,m+1]}]], {n,m+1,m+4}]] // Flatten

For n>1, A337411(n,k) = Sum_{j=1..2*n*(n-1)} T(n,j) * binomial(k,j).
T(n,k) = A338147(n,k) + A338148(n,k) = 2*A338147(n,k) - A338149(n,k) = 2*A338148(n,k) + A338149(n,k).
T(2,k) = A338142(2,k) = A325016(2,k) = A325008(2,k); T(3,k) = A338142(3,k).

cp's OEIS Frontend

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A325009 Triangle read by rows: T(n,k) is the number of unoriented colorings of the facets of a regular n-dimensional orthotope using exactly k colors. Row n has 2n columns.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A325016 Triangle read by rows: T(n,k) is the number of oriented colorings of the facets of a regular n-dimensional orthoplex using exactly k colors. Row n has 2^n columns.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A325002 Triangle read by rows: T(n,k) is the number of oriented colorings of the facets (or vertices) of a regular n-dimensional simplex using exactly k colors.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A325010 Triangle read by rows: T(n,k) is the number of chiral pairs of colorings of the facets of a regular n-dimensional orthotope using exactly k colors. Row n has 2n columns.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A338142 Triangle read by rows: T(n,k) is the number of oriented colorings of the edges of a regular n-D orthotope (or ridges of a regular n-D orthoplex) using exactly k colors. Row n has n*2^(n-1) columns.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A338146 Triangle read by rows: T(n,k) is the number of oriented colorings of the edges of a regular n-D orthoplex (or ridges of a regular n-D orthotope) using exactly k colors. Row 1 has 1 column; row n>1 has 2*n*(n-1) columns.

Views

A338146 Triangle read by rows: T(n,k) is the number of oriented colorings of the edges of a regular n-D orthoplex (or ridges of a regular n-D orthotope) using exactly k colors. Row 1 has 1 column; row n>1 has 2n(n-1) columns.