A325019 -id:A325019 - cp's OEIS frontend

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.

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).

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).

A325017 Triangle read by rows: T(n,k) is the number of unoriented 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, 1, 1, 4, 6, 3, 1, 20, 204, 1056, 2850, 4080, 2940, 840, 1, 400, 130899, 11230666, 347919225, 5158324560, 43174480650, 225086553300, 775894225050, 1831178115900, 3008073915000, 3439243962000, 2685727044000, 1366701336000, 408648240000, 54486432000

Offset: 1

Robert A. Russell, Jun 09 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 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 orthotope (cube) using exactly k colors.

			Triangle begins with T(1,1):
1  1
1  4   6    3
1 20 204 1056 2850 4080 2940 840
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.

Cf. A325016 (oriented), A325018 (chiral), A325019 (achiral), A325013 (up to k colors).
Other n-dimensional polytopes: A007318(n,k-1) (simplex), A325009 (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[#] + CI1[#]) pc[#]) & /@ IntegerPartitions[n]])/(n! 2^n)] /. CI[l_List] :> j^(Total[l][[2]])
array[n_, k_] := row[n] /. j -> k (* A325013 *)
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

A325013(n,k) = Sum_{j=1..2^n} T(n,j) * binomial(k,j).

T(n,k) = A325016(n,k) - A325018(n,k) = (A325016(n,k) + A325019(n,k)) / 2 = A325018(n,k) + A325019(n,k).

A325003 Triangle read by rows: T(n,k) is the number of achiral colorings of the facets (or vertices) of a regular n-dimensional simplex using exactly k colors.

Original entry on oeis.org

1, 0, 1, 2, 0, 1, 3, 3, 0, 1, 4, 6, 4, 0, 1, 5, 10, 10, 5, 0, 1, 6, 15, 20, 15, 6, 0, 1, 7, 21, 35, 35, 21, 7, 0, 1, 8, 28, 56, 70, 56, 28, 8, 0, 1, 9, 36, 84, 126, 126, 84, 36, 9, 0, 1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 0, 1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 0

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. For k <= n all the colorings are achiral.

The final zero in each row indicates no achiral colorings when each facet has a different color.

			Triangle begins with T(1,1):
1  0
1  2   0
1  3   3   0
1  4   6   4    0
1  5  10  10    5    0
1  6  15  20   15    6    0
1  7  21  35   35   21    7    0
1  8  28  56   70   56   28    8    0
1  9  36  84  126  126   84   36    9    0
1 10  45 120  210  252  210  120   45   10    0
1 11  55 165  330  462  462  330  165   55   11    0
1 12  66 220  495  792  924  792  495  220   66   12   0
1 13  78 286  715 1287 1716 1716 1287  715  286   78  13   0
1 14  91 364 1001 2002 3003 3432 3003 2002 1001  364  91  14  0
1 15 105 455 1365 3003 5005 6435 6435 5005 3003 1365 455 105 15 0
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. A325002 (oriented), A007318(n,k-1) (unoriented), A325001 (up to k colors).
Other n-dimensional polytopes: A325011 (orthotope), A325019 (orthoplex).
Cf. A198321.

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) = A325002(n,k) - 2*[k==n+1] = 2*A007318(n,k-1) - A325002(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).

A325018 Triangle read by rows: T(n,k) is the number of chiral pairs of 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

0, 1, 0, 0, 3, 3, 0, 1, 63, 662, 2400, 3900, 2940, 840, 0, 94, 97692, 10308758, 337560150, 5098740090, 42976836210, 224685801060, 775389028050, 1830791421900, 3007909258200, 3439214024400, 2685727044000, 1366701336000, 408648240000, 54486432000

Offset: 1

Robert A. Russell, Jun 09 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. 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 orthotope (cube) using exactly k colors.

			Triangle begins with T(1,1):
0 1
0 0  3   3
0 1 63 662 2400 3900 2940 840
For T(2,3)=3, each square has one of the three colors on two adjacent edges.

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.

Cf. A325016 (oriented), A325017 (unoriented), A325019 (achiral), A325014 (up to k colors).
Other n-dimensional polytopes: A325010 (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[#] - CI1[#]) pc[#]) & /@ IntegerPartitions[n]])/(n! 2^n)] /. CI[l_List] :> j^(Total[l][[2]])
array[n_, k_] := row[n] /. j -> k (* A325014 *)
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

A325014(n,k) = Sum_{j=1..2^n} T(n,j) * binomial(k,j).

T(n,k) = A325016(n,k) - A325017(n,k) = (A325016(n,k) - A325019(n,k)) / 2 = A325017(n,k) - A325019(n,k).

A338145 Triangle read by rows: T(n,k) is the number of achiral 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, 3, 0, 1, 68, 1200, 7268, 20025, 27750, 18900, 5040, 0, 0, 0, 0, 1, 93022, 293878020, 90807857080, 7503022894800, 258528829444320, 4681671089961600, 50981530073846400, 363246007692204000

Offset: 1

Robert A. Russell, Oct 12 2020

An achiral coloring is identical to its reflection. 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    3    0
  1 68 1200 7268 20025 27750 18900 5040 0 0 0 0
  ...
For T(2,2)=4, the achiral colorings are AAAB, AABB, ABAB, and ABBB. For T(2,3)=3, the achiral colorings are ABAC, ABCB, and ACBC.

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. A338142 (oriented), A338143 (unoriented), A338144 (chiral), A337410 (k or fewer colors), A325019 (orthotope vertices, orthoplex facets).

Cf. A327090 (simplex), A338149 (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}]], 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[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

A337410(n,k) = Sum_{j=1..n*2^(n-1)} T(n,j) * binomial(k,j).
T(n,k) = 2*A338143(n,k) - A338142(n,k) = A338142(n,k) - 2*A338144(n,k) = A338143(n,k) - A338144(n,k).
T(2,k) = A338149(2,k) = A325019(2,k) = A325011(2,k); T(3,k) = A338149(3,k).

A338149 Triangle read by rows: T(n,k) is the number of achiral 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, 3, 0, 1, 68, 1200, 7268, 20025, 27750, 18900, 5040, 0, 0, 0, 0, 1, 8198, 9055962, 1467050480, 74035775370, 1679679306420, 20864180531565, 159341117375160, 804216787965360, 2808560520334800, 6981656802951600

Offset: 1

Robert A. Russell, Oct 12 2020

An achiral coloring is identical to its reflection. 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       3          0
  1   68    1200       7268       20025         27750 18900 5040 0 0 0 0
  1 8198 9055962 1467050480 74035775370 1679679306420 ...
  ...
For T(2,2)=4, the achiral colorings are AAAB, AABB, ABAB, and ABBB. For T(2,3)=3, the 3 achiral colorings are ABAC, ABCB, and ACBC.

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. A338146 (oriented), A338147 (unoriented), A338148 (chiral), A337414 (k or fewer colors), A325011 (orthoplex vertices, orthotope facets).

Cf. A327090 (simplex), A338145 (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}]], 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
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, A337414(n,k) = Sum_{j=1..2*n*(n-1)} T(n,j) * binomial(k,j).
T(n,k) = 2*A338147(n,k) - A338146(n,k) = A338146(n,k) - 2*A338148(n,k) = A338147(n,k) - A338148(n,k).
T(2,k) = A338145(2,k) = A325019(2,k) = A325011(2,k); T(3,k) = A338145(3,k).

cp's OEIS Frontend

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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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.

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A325017 Triangle read by rows: T(n,k) is the number of unoriented colorings of the facets of a regular n-dimensional orthoplex using exactly k colors. Row n has 2^n columns.

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A325003 Triangle read by rows: T(n,k) is the number of achiral colorings of the facets (or vertices) of a regular n-dimensional simplex using exactly k colors.

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A325018 Triangle read by rows: T(n,k) is the number of chiral pairs of colorings of the facets of a regular n-dimensional orthoplex using exactly k colors. Row n has 2^n columns.

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A338145 Triangle read by rows: T(n,k) is the number of achiral 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.

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A338149 Triangle read by rows: T(n,k) is the number of achiral 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.

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A338149 Triangle read by rows: T(n,k) is the number of achiral 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.