A063843 -id:A063843 - cp's OEIS frontend

A331352 Number of chiral pairs of colorings of the edges (or triangular faces) of a regular 4-dimensional simplex with n available colors.

0, 6, 405, 7904, 76880, 486522, 2300305, 8806336, 28725192, 82626270, 214744629, 513368064, 1144198952, 2402617490, 4792612545, 9142333696, 16768783408, 29707141878, 51023629173, 85234690080, 138859666848

Offset: 1

Robert A. Russell, Jan 14 2020

A 4-dimensional simplex has 5 vertices and 10 edges. Its Schläfli symbol is {3,3,3}. The chiral colorings of its edges come in pairs, each the reflection of the other.

Index entries for linear recurrences with constant coefficients, signature (11, -55, 165, -330, 462, -462, 330, -165, 55, -11, 1).

Cf. A331350 (oriented), A063843 (unoriented), A331353 (achiral).
Other polychora: A331360 (8-cell), A331356 (16-cell), A338954 (24-cell), A338966 (120-cell, 600-cell).
Row 4 of A327085 (simplex edges and ridges) and A337885 (simplex faces and peaks).

Table[(24n^2 - 50n^3 + 20n^4 + 15n^6 - 10n^7 + n^10)/120, {n, 1, 25}]

a(n) = (24*n^2 - 50*n^3 + 20*n^4 + 15*n^6 - 10*n^7 + n^10) / 120.
a(n) = 6*C(n,2) + 387*C(n,3) + 6320*C(n,4) + 41350*C(n,5) + 135792*C(n,6) + 246540*C(n,7) + 252000*C(n,8) + 136080*C(n,9) + 30240*C(n,10), where the coefficient of C(n,k) is the number of colorings using exactly k colors.
a(n) = A331350(n) - A063843(n) = (A331350(n) - A331353(n)) / 2 = A063843(n) - A331353(n).

A337884 Array read by descending antidiagonals: T(n,k) is the number of unoriented colorings of the triangular faces of a regular n-dimensional simplex using k or fewer colors.

Original entry on oeis.org

1, 2, 1, 3, 5, 1, 4, 15, 34, 1, 5, 35, 792, 2136, 1, 6, 70, 10688, 4977909, 7013320, 1, 7, 126, 90005, 1533771392, 9930666709494, 1788782616656, 1, 8, 210, 533358, 132597435125, 234249157811872000, 12979877431438089379035, 53304527811667897248, 1

Offset: 2

Robert A. Russell, Sep 28 2020

Each chiral pair is counted as one when enumerating unoriented arrangements. An n-simplex has n+1 vertices. For n=2, the figure is a triangle with one triangular face. For n=3, the figure is a tetrahedron with 4 triangular faces. For higher n, the number of triangular faces is C(n+1,3).

Also the number of unoriented colorings of the peaks of a regular n-dimensional simplex. A peak of an n-simplex is an (n-3)-dimensional simplex.

			Table begins with T(2,1):
 1    2       3          4            5             6               7 ...
 1    5      15         35           70           126             210 ...
 1   34     792      10688        90005        533358         2437848 ...
 1 2136 4977909 1533771392 132597435125 5079767935320 110837593383153 ...
For T(3,4)=35, the 34 achiral arrangements are AAAA, AAAB, AAAC, AAAD, AABB, AABC, AABD, AACC, AACD, AADD, ABBB, ABBC, ABBD, ABCC, ABDD, ACCC, ACCD, ACDD, ADDD, BBBB, BBBC, BBBD, BBCC, BBCD, BBDD, BCCC, BCCD, BCDD, BDDD, CCCC, CCCD, CCDD, CDDD, and DDDD. The chiral pair is ABCD-ABDC.

E. M. Palmer and R. W. Robinson, Enumeration under two representations of the wreath product, Acta Math., 131 (1973), 123-143.

Cf. A337883 (oriented), A337885 (chiral), A337886 (achiral), A051168 (binary Lyndon words).
Other elements: A325000 (vertices), A327084 (edges).
Other polytopes: A337888 (orthotope), A337892 (orthoplex).
Rows 2-4 are A000027, A000332(n+3), A063843.

m=2; (* dimension of color element, here a triangular face *)
lw[n_,k_]:=lw[n, k]=DivisorSum[GCD[n,k],MoebiusMu[#]Binomial[n/#,k/#]&]/n (*A051168*)
cxx[{a_, b_},{c_, d_}]:={LCM[a, c], GCD[a, c] b d}
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)
combine[a : {{, } ...}, b : {{, } ...}] := Outer[cxx, a, b, 1]
CX[p_List, 0] := {{1, 1}} (* cycle index for partition p, m vertices *)
CX[{n_Integer}, m_] := If[2m>n, CX[{n}, n-m], CX[{n},m] = Table[{n/k, lw[n/k, m/k]}, {k, Reverse[Divisors[GCD[n, m]]]}]]
CX[p_List, m_Integer] := CX[p, m] = Module[{v = Total[p], q, r}, If[2 m > v, CX[p, v - m], q = Drop[p, -1]; r = Last[p]; compress[Flatten[Join[{{CX[q, m]}}, Table[combine[CX[q, m - j], CX[{r}, j]], {j, Min[m, r]}]], 2]]]]
pc[p_] := 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[pc[#] j^Total[CX[#, m+1]][[2]] & /@ IntegerPartitions[n+1]]/(n+1)!]
array[n_, k_] := row[n] /. j -> 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 vertices to a partition of n+1. It then determines the number of permutations for each partition and the cycle index for each partition using a formula for binary Lyndon words. If the value of m is increased, one can enumerate colorings of higher-dimensional elements beginning with T(m,1).

T(n,k) = A337883(n,k) - A337885(n,k) = (A337883(n,k) + A337886(n,k)) / 2 = A337885(n,k) + A337886(n,k).

A338953 Number of unoriented colorings of the 96 edges (or triangular faces) of the 4-D 24-cell using subsets of a set of n colors.

Original entry on oeis.org

1, 68774446639102959610154176, 5523164445430505961199114292414803649442426, 5448873034189827051954635848284422749083650351583379456, 10956401461402941741829572441752281718329313621842215239237500000

Offset: 1

Robert A. Russell, Nov 17 2020

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

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

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

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

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

A053400 Number of 3-multigraphs on n nodes.

Original entry on oeis.org

1, 4, 20, 276, 10688, 1601952, 892341888, 1799786093088, 13042490003160192, 341378170022783017472, 32526326484972756063585792, 11367103329997359707194173746176, 14669222110846093400698801891700529152

Offset: 1

Vladeta Jovovic, Jan 06 2000

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY,1973.

Andrew Howroyd, Table of n, a(n) for n = 1..50
Harald Fripertinger, The cycle type of the induced action on 2-subsets
Vladeta Jovovic, Formulae for the number T(n,k) of n-multigraphs on k nodes

Column k=3 of A063841.

Cf. A004102.

from itertools import combinations
from math import prod, gcd, factorial
from fractions import Fraction
from sympy.utilities.iterables import partitions
def A053400(n): return int(sum(Fraction(1<<(sum(p[r]*p[s]*gcd(r,s) for r,s in combinations(p.keys(),2))+sum((q>>1)*r+(q*r*(r-1)>>1) for q, r in p.items())<<1),prod(q**r*factorial(r) for q, r in p.items())) for p in partitions(n))) # Chai Wah Wu, Jul 09 2024

A063842 Number of colorings of K_4 using at most n colors.

Original entry on oeis.org

1, 11, 66, 276, 900, 2451, 5831, 12496, 24651, 45475, 79376, 132276, 211926, 328251, 493725, 723776, 1037221, 1456731, 2009326, 2726900, 3646776, 4812291, 6273411, 8087376, 10319375, 13043251, 16342236, 20309716, 25050026, 30679275, 37326201, 45133056

Offset: 0

N. J. A. Sloane, Aug 25 2001

a(n-1) is the number of unoriented ways to color the edges of a regular tetrahedron with up to n colors.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Vladeta Jovovic, Formulae for the number T(n,k) of n-multigraphs on k nodes
Marko R. Riedel, Counting multigraphs up to isomorphism
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

A row of A063841. Cf. A063843.

A327084(3,n) = a(n-1) (unoriented simplex edge colorings)

[1/24*(n^6+6*n^5+24*n^4+56*n^3+83*n^2+70*n+24): n in [0..35]]; // Vincenzo Librandi, Jul 21 2013

Needs["Combinatorica`"]
Table[Total[Table[CycleIndex[KSubsetGroup[GraphData[{4,k},"Automorphisms"],GraphData[{4,k},"EdgeIndices"]],s],{k,1,11}]]/.Table[s[i] -> n,{i,1,4}],{n,0,30}]  (* Geoffrey Critzer, Oct 22 2012 *)
CoefficientList[Series[(1 + 3 x + 7 x^2 + 3 x^3 + x^4) (1 + x) / (1 - x)^7, {x, 0, 35}], x] (* Vincenzo Librandi, Jul 21 2013 *)
LinearRecurrence[{7,-21,35,-35,21,-7,1},{1,11,66,276,900,2451,5831},40] (* Harvey P. Dale, Sep 10 2023 *)

a(n) = (1/4!)*(n^6 + 6*n^5 + 24*n^4 + 56*n^3 + 83*n^2 + 70*n + 24).

G.f.: (1 + 3*x + 7*x^2 + 3*x^3 + x^4)*(1+x)/(1-x)^7. - M. F. Hasler, Jan 19 2012

More terms from Vladeta Jovovic, Sep 02 2001

A338957 Number of unoriented colorings of the 96 edges (or triangular faces) of the 4-D 24-cell using exactly n colors.

Original entry on oeis.org

1, 68774446639102959610154174, 5523164445430505754875774375105924818979901, 5448873034167734394172913824852272971748608894646534804, 10956401434158576570935668826433407535831446552957081921713485225

Offset: 1

Robert A. Russell, Nov 17 2020

Each chiral pair is counted as one when enumerating unoriented arrangements. The Schläfli symbol of the 24-cell is {3,4,3}. It has 24 octahedral facets. It is self-dual. For n>96, a(n) = 0.

Robert A. Russell, Table of n, a(n) for n = 1..96

Cf. A338956 (oriented), A338958 (chiral), A338959 (achiral), A338953 (up to n colors), A338949 (vertices, facets), A063843 (5-cell), A331359 (8-cell edges, 16-cell faces), A331355 (16-cell edges, 8-cell faces), A338981 (120-cell, 600-cell).

bp[j_] := Sum[k! StirlingS2[j, k] x^k, {k, 0, j}] (* binomial series *)
Drop[CoefficientList[bp[8]/12+bp[12]/8+bp[16]/8+bp[18]/9+bp[20]/6+19bp[24]/96+bp[32]/24+bp[36]/36+43bp[48]/1152+bp[50]/16+bp[52]/96+bp[60]/96+bp[96]/1152,x],1]

A338953(n) = Sum_{j=1..Min(n,96)} a(n) * binomial(n,j).

a(n) = A338956(n) - A338958(n) = (A338956(n) + A338959(n)) / 2 = A338958(n) + A338959(n).

A053420 Number of 4-multigraphs on n nodes.

Original entry on oeis.org

1, 5, 35, 900, 90005, 43571400, 95277592625, 925609100039625, 40119721052610123750, 7833164300852979350336250, 6953552738579427778531249187500, 28293472829338822230349054996265275000, 531350037528849507720092485196308155336875000

Offset: 1

Vladeta Jovovic, Jan 11 2000

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973.

Andrew Howroyd, Table of n, a(n) for n = 1..50
Harald Fripertinger, The cycle type of the induced action on 2-subsets
Vladeta Jovovic, Formulae for the number T(n,k) of n-multigraphs on k nodes

Column k=4 of A063841.

Cf. A000088, A004102, A053400.

from itertools import combinations
from math import prod, gcd, factorial
from fractions import Fraction
from sympy.utilities.iterables import partitions
def A053420(n): return int(sum(Fraction(5**(sum(p[r]*p[s]*gcd(r,s) for r,s in combinations(p.keys(),2))+sum((q>>1)*r+(q*r*(r-1)>>1) for q, r in p.items())),prod(q**r*factorial(r) for q, r in p.items())) for p in partitions(n))) # Chai Wah Wu, Jul 09 2024

Terms a(12) and beyond from Andrew Howroyd, Oct 22 2017

A053421 Number of 5-multigraphs on n nodes.

Original entry on oeis.org

1, 6, 56, 2451, 533358, 661452084, 4364646955812, 152397092027960154, 28427450083725134688228, 28645398830642924774967347088, 157458251108667629202718200130101672, 4760428376101385226312810920945121043818096

Offset: 1

Vladeta Jovovic, Jan 11 2000

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973.

Andrew Howroyd, Table of n, a(n) for n = 1..50
Harald Fripertinger, The cycle type of the induced action on 2-subsets
Vladeta Jovovic, Formulae for the number T(n,k) of n-multigraphs on k nodes

Column k=5 of A063841.

Cf. A000088, A004102, A053400.

from itertools import combinations
from math import prod, gcd, factorial
from fractions import Fraction
from sympy.utilities.iterables import partitions
def A053421(n): return int(sum(Fraction(6**(sum(p[r]*p[s]*gcd(r,s) for r,s in combinations(p.keys(),2))+sum((q>>1)*r+(q*r*(r-1)>>1) for q, r in p.items())),prod(q**r*factorial(r) for q, r in p.items())) for p in partitions(n))) # Chai Wah Wu, Jul 09 2024

a(12) from Andrew Howroyd, Oct 22 2017

A270842 Number of nonisomorphic edge colorings of the Petersen graph with at most n colors.

Original entry on oeis.org

1, 396, 123786, 9002912, 254721400, 3920311044, 39571426713, 293231076608, 1715840171595, 8333541708700, 34810892718492, 128392921513440, 426551317876970, 1296405100924948, 3649123762524675, 9607693522053120, 23853550135649477, 56222046462953772

Offset: 1

Marko Riedel, Mar 24 2016

These are not colorings in the strict sense, since there is no requirement that adjacent edges have different colors. - N. J. A. Sloane, Mar 28 2016

Colin Barker, Table of n, a(n) for n = 1..1000
Math StackExchange, Edge colorings of the Petersen graph

Cf. A270843, A063843. See A159233 for edge colorings where adjacent edges must have different colors.

a(n) = n^15/120 + 5*n^9/24 + 5*n^5/12 + 11*n^3/30; \\ Altug Alkan, Mar 25 2016

Cycle index of the automorphisms acting on the edges is (1/120)*S[1]^15+(5/24)*S[2]^6*S[1]^3+(1/4)*S[4]^3*S[2]*S[1]+(1/6)*S[3]^5+(1/6)*S[3]*S[6]^2+(1/5)*S[5]^3.
a(n) = n^15/120 + 5*n^9/24 + 5*n^5/12 + 11*n^3/30.
G.f.: x*(1 + 380*x + 117570*x^2 + 7069296*x^3 + 125309188*x^4 + 856514276*x^5 + 2594956089*x^6 + 3729352800*x^7 + 2594956089*x^8 + 856514276*x^9 + 125309188*x^10 + 7069296*x^11 + 117570*x^12 + 380*x^13 + x^14) / (1 - x)^16. - Colin Barker, Dec 24 2017

A270843 Number of nonisomorphic edge colorings of the Petersen graph with exactly n colors.

Original entry on oeis.org

1, 394, 122601, 8510140, 210940745, 2524556538, 17167621086, 72787256640, 202996629360, 382918536000, 492133561920, 424994169600, 236107872000, 76281004800, 10897286400

Offset: 1

Marko Riedel, Mar 24 2016

This is zero when n is more than fifteen because only fifteen edges are available.
These are not colorings in the strict sense, since there is no requirement that adjacent edges have different colors. - N. J. A. Sloane, Mar 28 2016
The value for n=15 is 15!/120 because all orbits are the same size namely 120 (order of the symmetric group on five elements) when each of the 15 edges has a unique color. - Marko Riedel, Mar 28 2016

Math StackExchange, Edge colorings of the Petersen graph

Cf. A270842, A063843.

Cycle index of the automorphisms acting on the edges is (1/120)*S[1]^15+(5/24)*S[2]^6*S[1]^3+(1/4)*S[4]^3*S[2]*S[1]+(1/6)*S[3]^5+(1/6)*S[3]*S[6]^2+(1/5)*S[5]^3.

Inclusion-exclusion yields a(n) = sum(C(n, q)*(-1)^q*A270842(n - q), q = 0 .. n)

cp's OEIS Frontend

A331352 Number of chiral pairs of colorings of the edges (or triangular faces) of a regular 4-dimensional simplex with n available colors.

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A337884 Array read by descending antidiagonals: T(n,k) is the number of unoriented colorings of the triangular faces of a regular n-dimensional simplex using k or fewer colors.

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A338953 Number of unoriented colorings of the 96 edges (or triangular faces) of the 4-D 24-cell using subsets of a set of n colors.

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A053400 Number of 3-multigraphs on n nodes.

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A063842 Number of colorings of K_4 using at most n colors.

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A338957 Number of unoriented colorings of the 96 edges (or triangular faces) of the 4-D 24-cell using exactly n colors.

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A053420 Number of 4-multigraphs on n nodes.

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A053421 Number of 5-multigraphs on n nodes.

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