A046023 -id:A046023 - cp's OEIS frontend

A060530 Number of inequivalent ways to color edges of a cube using at most n colors.

0, 1, 218, 22815, 703760, 10194250, 90775566, 576941778, 2863870080, 11769161895, 41669295250, 130772947481, 371513523888, 970769847320, 2362273657030, 5406141568500, 11728193258496, 24276032182173, 48201464902410, 92221684354915

Offset: 0

N. J. A. Sloane, Apr 11 2001

Here inequivalent means under the action of the rotation group of the cube, of order 24, which in its action on the edges has cycle index (x1^12 + 3*x2^6 + 6*x4^3 + 6*x1^2*x2^5 + 8*x3^4)/24.
Also, number of inequivalent colorings of the edges of a regular octahedron using at most n colors. - José H. Nieto S., Jan 19 2012
From Robert A. Russell, Oct 08 2020: (Start)
Each chiral pair is counted as two when enumerating oriented arrangements. The Schläfli symbols for the regular octahedron and cube are {3,4} and {4,3} respectively. They are mutually dual.
There are 24 elements in the rotation group of the regular octahedron/cube. They divide into five conjugacy classes. The first formula is obtained by averaging the edge cycle indices after replacing x_i^j with n^j according to the Pólya enumeration theorem.
  Conjugacy Class    Count    Even Cycle Indices
  Identity              1     x_1^12
  Vertex rotation       8     x_3^4
  Edge rotation         6     x_1^2x_2^5
  Small face rotation   6     x_4^3
  Large face rotation   3     x_2^6 (End)
Also, number of ways of coloring the vertices of the truncated tetrahedron or faces of the triakis tetrahedron up to rotation and reflection. - Peter Kagey, Nov 27 2024

N. G. De Bruijn, Polya's theory of counting, in E. F. Beckenbach, ed., Applied Combinatorial Mathematics, Wiley, 1964, pp. 144-184 (see p. 147).

Harry J. Smith, Table of n, a(n) for n=0..200
Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Cf. A199406 (unoriented), A337406 (chiral), A331351 (achiral).
Other elements: A000543 (cube vertices, octahedron faces), A047780 (cube faces, octahedron vertices).
Cf. A046023 (tetrahedron), A282670 (dodecahedron/icosahedron).
Row 3 of A337407 (orthotope edges, orthoplex ridges) and A337411 (orthoplex edges, orthotope ridges).

Table[(n^12+6n^7+3n^6+8n^4+6n^3)/24,{n,0,20}] (* Harvey P. Dale, Feb 13 2013 *)

{ for (n=0, 200, write("b060530.txt", n, " ", (n^12 + 6*n^7 + 3*n^6 + 8*n^4 + 6*n^3)/24); ) } \\ Harry J. Smith, Jul 06 2009

a(n) = (n^12 + 6*n^7 + 3*n^6 + 8*n^4 + 6*n^3)/24. (Replace all x_i's in the cycle index by n.)
G.f.: -x*(150*x^10 +19758*x^9 +425032*x^8 +2763481*x^7 +6769435*x^6 +6773089*x^5 +2763307*x^4 +423883*x^3 +20059*x^2 +205*x +1)/(x -1)^13. - Colin Barker, Aug 13 2012
From Robert A. Russell, Oct 08 2020: (Start)
a(n) = 1*C(n,1) + 216*C(n,2) + 22164*C(n,3) + 613804*C(n,4) + 6901425*C(n,5) + 39713430*C(n,6) + 131754420*C(n,7) + 267165360*C(n,8) + 336798000*C(n,9) + 257796000*C(n,10) + 109771200*C(n,11) + 19958400*C(n,12), where the coefficient of C(n,k) is the number of oriented colorings using exactly k colors.
a(n) = A199406(n) + A337406(n) = 2*A199406(n) - A331351(n) = 2*A337406(n) + A331351(n). (End)

Entry revised by N. J. A. Sloane, Jan 03 2005

A327083 Array read by descending antidiagonals: A(n,k) is the number of oriented colorings of the edges of a regular n-dimensional simplex using up to k colors.

Original entry on oeis.org

1, 2, 1, 3, 4, 1, 4, 11, 12, 1, 5, 24, 87, 40, 1, 6, 45, 416, 1197, 184, 1, 7, 76, 1475, 18592, 42660, 1296, 1, 8, 119, 4236, 166885, 3017600, 4223313, 17072, 1, 9, 176, 10437, 1019880, 85025050, 1748176768, 1139277096, 424992

Offset: 1

Robert A. Russell, Aug 19 2019

An n-dimensional simplex has n+1 vertices and (n+1)*n/2 edges. For n=1, the figure is a line segment with one edge. For n-2, the figure is a triangle with three edges. For n=3, the figure is a tetrahedron with six edges. The Schläfli symbol, {3,...,3}, of the regular n-dimensional simplex consists of n-1 threes. Two oriented colorings are the same if one is a rotation of the other; chiral pairs are counted as two.

A(n,k) is also the number of oriented colorings of (n-2)-dimensional regular simplices in an n-dimensional simplex using up to k colors. Thus, A(2,k) is also the number of oriented colorings of the vertices (0-dimensional simplices) of an equilateral triangle.

			Array begins with A(1,1):
  1  2    3     4      5       6       7        8        9        10 ...
  1  4   11    24     45      76     119      176      249       340 ...
  1 12   87   416   1475    4236   10437    22912    45981     85900 ...
  1 40 1197 18592 166885 1019880 4738153 17962624 58248153 166920040 ...
  ...
For A(2,3) = 11, the nine achiral colorings are AAA, AAB, AAC, ABB, ACC, BBB, BBC, BCC, and CCC. The chiral pair is ABC-ACB.

Robert A. Russell, Table of n, a(n) for n = 1..325 First 25 antidiagonals.
Harald Fripertinger, The cycle type of the induced action on 2-subsets
E. M. Palmer and R. W. Robinson, Enumeration under two representations of the wreath product, Acta Math., 131 (1973), 123-143.

Cf. A327084 (unoriented), A327085 (chiral), A327086 (achiral), A327087 (exactly k colors), A324999 (vertices, facets), A337883 (faces, peaks), A337407 (orthotope edges, orthoplex ridges), A337411 (orthoplex edges, orthotope ridges).
Rows 1-4 are A000027, A006527, A046023, A331350.
Column 2 is A218144(n+1).

CycleX[{2}] = {{1,1}}; (* cycle index for permutation with given cycle structure *)
CycleX[{n_Integer}] := CycleX[n] = If[EvenQ[n], {{n/2,1}, {n,(n-2)/2}}, {{n,(n-1)/2}}]
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)
CycleX[p_List] := CycleX[p] = compress[Join[CycleX[Drop[p,-1]], If[Last[p] > 1, CycleX[{Last[p]}], ## &[]], If[# == Last[p], {#, Last[p]}, {LCM[#, Last[p]], GCD[#, Last[p]]}] & /@ Drop[p,-1]]]
pc[p_List] := Module[{ci, mb}, mb = DeleteDuplicates[p]; ci = Count[p, #] & /@ mb; Total[p]!/(Times @@ (ci!) Times @@ (mb^ci))] (*partition count*)
row[n_Integer] := row[n] = Factor[Total[If[EvenQ[Total[1-Mod[#,2]]], pc[#] j^Total[CycleX[#]][[2]], 0] & /@ IntegerPartitions[n+1]]/((n+1)!/2)]
array[n_, k_] := row[n] /. j -> k
Table[array[n,d-n+1], {d,1,10}, {n,1,d}] // Flatten
(* Using Fripertinger's exponent per Andrew Howroyd's code in A063841: *)
pc[p_] := Module[{ci, mb}, mb = DeleteDuplicates[p]; ci = Count[p, #] &/@ mb; Total[p]!/(Times @@ (ci!) Times @@ (mb^ci))]
ex[v_] := Sum[GCD[v[[i]], v[[j]]], {i,2,Length[v]}, {j,i-1}] + Total[Quotient[v,2]]
array[n_,k_] := Total[If[EvenQ[Total[1-Mod[#,2]]], pc[#]k^ex[#], 0] &/@ IntegerPartitions[n+1]]/((n+1)!/2)
Table[array[n,d-n+1], {d,10}, {n,d}] // Flatten

The algorithm used in the Mathematica program below assigns each permutation of the vertices to a partition of n+1. It then determines the number of permutations for each partition and the cycle index for each partition.
A(n,k) = Sum_{j=1..(n+1)*n/2} A327087(n,j) * binomial(k,j).
A(n,k) = A327084(n,k) + A327085(n,k) = 2*A327084(n,k) - A327086(n,k) = 2*A327085(n,k) + A327086(n,k).

A282670 Number of inequivalent ways to color the edges of a dodecahedron using at most n colors.

Original entry on oeis.org

0, 1, 17912448, 3431529649899, 19215359484207104, 15522042948408209375, 3684565329384186949248, 375655671519845961645597, 20632333988160040350515200, 706519304587399981447927557, 16666666666669166670000400000, 290823371148118276083759139095

Offset: 0

David Nacin, Feb 20 2017

Cycle index of symmetry group A5 acting on the 30 edges of the dodecahedron is (24s(5)^6 + 20s(3)^10 + 15s(2)^14*s(1)^2 + s(1)^30)/60.
Also the number of inequivalent ways to color the edges of the icosahedron using at most n colors.
From Robert A. Russell, Oct 03 2020: (Start)
Each chiral pair is counted as two when enumerating oriented arrangements. The Schläfli symbols for the regular icosahedron and regular dodecahedron are {3,5} and {5,3} respectively. They are mutually dual. There are 60 elements in the rotation group of the regular dodecahedron/icosahedron. They divide into five conjugacy classes. The first formula is obtained by averaging the edge cycle indices after replacing x_i^j with n^j according to the Pólya enumeration theorem.
  Conjugacy Class    Count    Even Cycle Indices
  Identity              1     x_1^30
  Edge rotation        15     x_1^2x_2^14
  Vertex rotation      20     x_3^10
  Small face rotation  12     x_5^6
Large face rotation  12     x_5^6  (End)

			There are a(2) = 17912448 inequivalent ways to color the edges of the dodecahedron using at most two colors.

Index entries for linear recurrences with constant coefficients, signature (31, -465, 4495, -31465, 169911, -736281, 2629575, -7888725, 20160075, -44352165, 84672315, -141120525, 206253075, -265182525, 300540195, -300540195, 265182525, -206253075, 141120525, -84672315, 44352165, -20160075, 7888725, -2629575, 736281, -169911, 31465, -4495, 465, -31, 1).

Other elements: A054472 (dodecahedron vertices, icosahedron faces), A000545 (dodecahedron faces, icosahedron vertices).
Other polyhedra: A046023 (tetrahedron), A060530 (cube/octahedron).
Cf. A337963 (unoriented), A337964 (chiral), A337953 (achiral).

Table[(24n^6+20n^10+15n^16+n^30)/60, {n, 0, 16}]

a(n) = n^6 (n^24 + 15 n^10 + 20 n^4 + 24)/60.
G.f.: x*(1 + x)*(1 + 17912416*x + 3430956452060*x^2 + 19105559437892000*x^3 + 14908856825730677891*x^4 + 3197392859155796794496*x^5 + 265368238349945588707496*x^6 + 10365795256050146806088576*x^7 + 215154060506484358838662001*x^8 + 2568188846096433625477331936*x^9 + 18582986600475456162494990756*x^10 + 84400699070086923625163495456*x^11 + 245956255494355672481225103371*x^12 + 465612713610802763378946154496*x^13 + 575747234318647571242943474096*x^14 + 465612713610802763378946154496*x^15 + 245956255494355672481225103371*x^16 + 84400699070086923625163495456*x^17 + 18582986600475456162494990756*x^18 + 2568188846096433625477331936*x^19 + 215154060506484358838662001*x^20 + 10365795256050146806088576*x^21 + 265368238349945588707496*x^22 + 3197392859155796794496*x^23 + 14908856825730677891*x^24 + 19105559437892000*x^25 + 3430956452060*x^26 + 17912416*x^27 + x^28) / (1 - x)^31. - Colin Barker, Mar 30 2019
From Robert A. Russell, Oct 03 2020: (Start)
a(n) = 1*C(n,1) + 17912446*C(n,2) + 3431475912558*C(n,3) + 19201633473082192*C(n,4) + 15426000466104548370*C(n,5) + 3591721233455676488292*C(n,6) + 350189004698594439734160*C(n,7) + 17729388555701917767855840*C(n,8) + 534044352737570253478824960*C(n,9) + 10485619820879148545218980480*C(n,10) + 143066535726280748444739676800*C(n,11) + 1420876074163106703694904352000*C(n,12) + 10631861498419617103267350931200*C(n,13) + 61515486939441778743810979468800*C(n,14) + 280711222366395106969585943040000*C(n,15) + 1025499893865270227589218761728000*C(n,16) + 3032858772294885663526454593536000*C(n,17) + 7319173455487770465200322686976000*C(n,18) + 14487618384525410959295952691200000*C(n,19) + 23580333216029318427870396825600000*C(n,20) + 31555723729541430372276884520960000*C(n,21) + 34619561317726617824610327429120000*C(n,22) + 30946535969611314628728933580800000*C(n,23) + 22311118596400512968549479219200000*C(n,24) + 12771433990957347267674112000000000*C(n,25) + 5668281691036644651075462758400000*C(n,26) + 1879979643918904128084836352000000*C(n,27) + 438404032189593555246120960000000*C(n,28) + 64102774454612839170441216000000*C(n,29) + 4420880996869850977271808000000*C(n,30), where the coefficient of C(n,k) is the number of oriented colorings using exactly k colors.
a(n) = A337963(n) + A337964(n) = 2*A337963(n) - A337953(n) = 2*A337964(n) + A337953(n). (End)

A282820 Number of inequivalent ways to color the edges of a tetrahedron using at most n colors so that no color appears more than twice.

Original entry on oeis.org

0, 0, 0, 9, 132, 720, 2580, 7245, 17304, 36792, 71640, 130185, 223740, 367224, 579852, 885885, 1315440, 1905360, 2700144, 3752937, 5126580, 6894720, 9142980, 11970189, 15489672, 19830600, 25139400, 31581225, 39341484, 48627432, 59669820, 72724605, 88074720

Offset: 0

David Nacin, Feb 22 2017

			For n = 3 we get a(3) = 9 ways to color the edges of a tetrahedron in three colors so that no color appears more than twice.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A282817, A282818, A282819, A046023 (tetrahedral edge colorings without restriction).

Table[(n-2)*n*(n-1)*(n^3+3*n^2-10*n-6)/12, {n, 0, 32}]

a(n) = (n-2)*n*(n-1)*(n^3+3*n^2-10*n-6)/12 \\ Charles R Greathouse IV, Feb 22 2017

concat(vector(3), Vec(3*x^3*(3 - x)*(1 + 8*x + x^2) / (1 - x)^7 + O(x^40))) \\ Colin Barker, Feb 22 2017

a(n) = (n-2)*n*(n-1)*(n^3+3*n^2-10*n-6)/12.
From Colin Barker, Feb 22 2017: (Start)
G.f.: 3*x^3*(3 - x)*(1 + 8*x + x^2) / (1 - x)^7.
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n>6. (End)

A282818 Number of inequivalent ways to color the edges of a tetrahedron using at most n colors so that no two adjacent edges have the same color.

Original entry on oeis.org

0, 0, 0, 2, 20, 110, 460, 1540, 4312, 10500, 22920, 45870, 85580, 150722, 252980, 407680, 634480, 958120, 1409232, 2025210, 2851140, 3940790, 5357660, 7176092, 9482440, 12376300, 15971800, 20398950, 25805052, 32356170, 40238660, 49660760, 60854240, 74076112

Offset: 0

David Nacin, Feb 22 2017

			For n = 3 we get a(3) = 2 distinct ways to color the edges of a tetrahedron with three colors so that no two adjacent edges have the same color.

Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1)

Cf. A282819, A282820, A046023 (tetrahedral edge colorings without restriction).

Table[n (n - 1) (n - 2) (n^3 - 9 n^2 + 32 n - 38)/12, {n, 0, 34}]

a(n) = n*(n-1)*(n-2)*(n^3-9*n^2+32*n-38)/12 \\ Charles R Greathouse IV, Feb 22 2017

a(n) = n*(n-1)*(n-2)*(n^3-9*n^2+32*n-38)/12.
G.f.: -2*x^3*(1+3*x+6*x^2+20*x^3)/(x-1)^7 . - R. J. Mathar, Feb 23 2017
a(n) = 2*A249460(n). - R. J. Mathar, Feb 23 2017

A282819 Number of inequivalent ways to color the edges of a tetrahedron using at most n colors so that no two opposite edges have the same color.

Original entry on oeis.org

0, 0, 2, 22, 152, 680, 2270, 6202, 14672, 31152, 60810, 110990, 191752, 316472, 502502, 771890, 1152160, 1677152, 2387922, 3333702, 4572920, 6174280, 8217902, 10796522, 14016752, 18000400, 22885850, 28829502, 36007272, 44616152, 54875830, 67030370, 81349952

Offset: 0

David Nacin, Feb 22 2017

			For n = 2 we get a(2) = 2 distinct ways to color the edges of a tetrahedron in two colors so that no two opposite edges have the same color.

Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A282816, A282818, A282820. A046023 (tetrahedral edge colorings without restriction).

Table[(n - 1) n (n^4 - 2 n^3 + n^2 + 8)/12, {n, 0, 33}]

a(n) = n*(n-1)*(n^4-2*n^3+n^2+8)/12 \\ Charles R Greathouse IV, Feb 22 2017

a(n) = n*(n-1)*(n^4-2*n^3+n^2+8)/12.
G.f.: -2*x^2*(1+4*x+20*x^2+4*x^3+x^4) / (x-1)^7 . - R. J. Mathar, Feb 23 2017
a(n) = 2*A282816(n). - R. J. Mathar, Feb 23 2017

A337899 Number of chiral pairs of colorings of the edges of a regular tetrahedron using n or fewer colors.

Original entry on oeis.org

0, 1, 21, 140, 575, 1785, 4606, 10416, 21330, 40425, 71995, 121836, 197561, 308945, 468300, 690880, 995316, 1404081, 1943985, 2646700, 3549315, 4694921, 6133226, 7921200, 10123750, 12814425, 16076151, 20001996

Offset: 1

Robert A. Russell, Sep 28 2020

nonn

Each member of a chiral pair is a reflection, but not a rotation, of the other. A regular tetrahedron has 6 edges and Schläfli symbol {3,3}.

			For a(2)=1, two opposite edges and one edge connecting those have one color; the other three edges have the other color.

Index entries for linear recurrences with constant coefficients, signature (7, -21, 35, -35, 21, -7, 1).

Cf. A046023(unoriented), A063842(n-1) (oriented), A037270 (chiral).
Other elements: A000332 (vertices and faces).
Other polyhedra: A337406 (cube/octahedron).
Row 3 of A327085 (chiral pairs of colorings of edges or ridges of an n-simplex).

Mathematica
```
Table[(n-1)n^2(n+1)(n^2-2)/24, {n, 40}]
```

a(n) = (n-1) * n^2 * (n+1) * (n^2-2) / 24.
a(n) = 1*C(n,2) + 18*C(n,3) + 62*C(n,4) + 75*C(n,5) + 30*C(n,6), where the coefficient of C(n,k) is the number of chiral pairs of colorings using exactly k colors.
a(n) = A046023(n) - A063842(n-1) = (A046023(n) - A037270(n)) / 2 = A063842(n-1) - A037270(n).
G.f.: x^2 * (1+x) * (1+13x+x^2)/(1-x)^7.

cp's OEIS Frontend

A060530 Number of inequivalent ways to color edges of a cube using at most n colors.

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A327083 Array read by descending antidiagonals: A(n,k) is the number of oriented colorings of the edges of a regular n-dimensional simplex using up to k colors.

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A282670 Number of inequivalent ways to color the edges of a dodecahedron using at most n colors.

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A282820 Number of inequivalent ways to color the edges of a tetrahedron using at most n colors so that no color appears more than twice.

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A282818 Number of inequivalent ways to color the edges of a tetrahedron using at most n colors so that no two adjacent edges have the same color.

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A282819 Number of inequivalent ways to color the edges of a tetrahedron using at most n colors so that no two opposite edges have the same color.

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A337899 Number of chiral pairs of colorings of the edges of a regular tetrahedron using n or fewer colors.

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