cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A368983 Number of connected graphs with loops (symmetric relations) on n unlabeled vertices with n edges.

Original entry on oeis.org

1, 1, 1, 3, 6, 14, 33, 81, 204, 526, 1376, 3648, 9792, 26485, 72233, 198192, 546846, 1515687, 4218564, 11782427, 33013541, 92759384, 261290682, 737688946, 2086993034, 5915398230, 16795618221, 47763406249, 136028420723, 387928330677, 1107692471888, 3166613486137
Offset: 0

Views

Author

Andrew Howroyd, Jan 11 2024

Keywords

Comments

The graphs considered here can have loops but not parallel edges.

Examples

			Representatives of the a(3) = 3 graphs are:
   {{1,2}, {1,3}, {2,3}},
   {{1}, {1,2}, {1,3}},
   {{1}, {1,2}, {2,3}}.

Links

Crossrefs

A diagonal of A322114.
The labeled version is A368951.
Cf. A000081, A001429, A027852, A068051, A283755, A368984 (not necessarily connected).

Programs

  • PARI
    \\ TreeGf gives gf of A000081.
TreeGf(N)={my(A=vector(N, j, 1)); for (n=1, N-1, A[n+1] = 1/n * sum(k=1, n, sumdiv(k, d, d*A[d]) * A[n-k+1] ) ); x*Ser(A)}
seq(n)={my(t=TreeGf(n)); my(g(e)=subst(t + O(x*x^(n\e)), x, x^e) + O(x*x^n)); Vec(1 + (sum(d=1, n, eulerphi(d)/d*log(1/(1-g(d)))) + ((1+g(1))^2/(1-g(2))-1)/2 - (g(1)^2 + g(2)))/2)}

Formula

a(n) = A000081(n) + A001429(n) = A068051(n) - A027852(n) for n > 0.

A369147 Number of unlabeled loop-graphs covering n vertices such that it is not possible to choose a different vertex from each edge (non-choosable).

Original entry on oeis.org

0, 0, 1, 7, 52, 411, 4440, 73886, 2128608, 111533208, 10812478194, 1945437194308, 650378721118910, 404749938336301313, 470163239887698682289, 1022592854829028310302180, 4177826139658552046624979658, 32163829440870460348768017832607, 468021728889827507080865185809438918
Offset: 0

Views

Author

Gus Wiseman, Jan 23 2024

Keywords

Examples

			The a(0) = 0 through a(3) = 7 loop-graphs (loops shown as singletons):
  .  .  {{1},{2},{1,2}}  {{1},{2},{3},{1,2}}
                         {{1},{2},{1,2},{1,3}}
                         {{1},{2},{1,3},{2,3}}
                         {{1},{1,2},{1,3},{2,3}}
                         {{1},{2},{3},{1,2},{1,3}}
                         {{1},{2},{1,2},{1,3},{2,3}}
                         {{1},{2},{3},{1,2},{1,3},{2,3}}

Links

Crossrefs

Without the choice condition we have A322700, labeled A322661.
The complement for exactly n edges is A368984, labeled A333331 (maybe).
The labeled complement is A369140, covering case of A368927.
The labeled version is A369142, covering case of A369141.
This is the covering case of A369146.
The complement is counted by A369200, covering case of A369145.
Without loops we have A369202, covering case of A140637.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A000666 counts unlabeled loop-graphs, labeled A006125 (shifted left).
A002494 counts unlabeled covering graphs, labeled A006129.
A007716 counts non-isomorphic multiset partitions, connected A007718.
Cf. A000088, A000612, A005703, A055621, A062740, A134964, A137917, A140638, A367868, A368835, A369199.

Programs

  • Mathematica
    brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{(Union@@m)[[i]],p[[i]]}, {i,Length[p]}])],{p,Permutations[Range[Length[Union@@m]]]}]]];
Table[Length[Union[brute /@ Select[Subsets[Subsets[Range[n],{1,2}]], Union@@#==Range[n] && Length[Select[Tuples[#],UnsameQ@@#&]]==0&]]],{n,0,4}]

Formula

First differences of A369146.
a(n) = A322700(n) - A369200(n). - Andrew Howroyd, Feb 02 2024

Extensions

a(6) onwards from Andrew Howroyd, Feb 02 2024

A369200 Number of unlabeled loop-graphs covering n vertices such that it is possible to choose a different vertex from each edge (choosable).

Original entry on oeis.org

1, 1, 3, 7, 18, 43, 112, 282, 740, 1940, 5182, 13916, 37826, 103391, 284815, 788636, 2195414, 6137025, 17223354, 48495640, 136961527, 387819558, 1100757411, 3130895452, 8922294498, 25470279123, 72823983735, 208515456498, 597824919725, 1716072103910, 4931540188084
Offset: 0

Views

Author

Gus Wiseman, Jan 23 2024

Keywords

Comments

These are covering loop-graphs with at most one cycle (unicyclic) in each connected component.

Examples

			Representatives of the a(1) = 1 through a(4) = 18 loop-graphs (loops shown as singletons):
  {{1}}  {{1,2}}      {{1},{2,3}}          {{1,2},{3,4}}
         {{1},{2}}    {{1,2},{1,3}}        {{1},{2},{3,4}}
         {{1},{1,2}}  {{1},{2},{3}}        {{1},{1,2},{3,4}}
                      {{1},{2},{1,3}}      {{1},{2,3},{2,4}}
                      {{1},{1,2},{1,3}}    {{1},{2},{3},{4}}
                      {{1},{1,2},{2,3}}    {{1,2},{1,3},{1,4}}
                      {{1,2},{1,3},{2,3}}  {{1,2},{1,3},{2,4}}
                                           {{1},{2},{3},{1,4}}
                                           {{1},{2},{1,3},{1,4}}
                                           {{1},{2},{1,3},{2,4}}
                                           {{1},{2},{1,3},{3,4}}
                                           {{1},{1,2},{1,3},{1,4}}
                                           {{1},{1,2},{1,3},{2,4}}
                                           {{1},{1,2},{2,3},{2,4}}
                                           {{1},{1,2},{2,3},{3,4}}
                                           {{1},{2,3},{2,4},{3,4}}
                                           {{1,2},{1,3},{1,4},{2,3}}
                                           {{1,2},{1,3},{2,4},{3,4}}

Links

Crossrefs

Without the choice condition we have A322700, labeled A322661.
Without loops we have A368834, covering case of A134964.
For exactly n edges we have A368984, labeled A333331 (maybe).
The labeled version is A369140, covering case of A368927.
The labeled complement is A369142, covering case of A369141.
This is the covering case of A369145.
The complement is counted by A369147, covering case of A369146.
The complement without loops is A369202, covering case of A140637.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A000666 counts unlabeled loop-graphs, labeled A006125 (shifted left).
A006129 counts covering graphs, unlabeled A002494.
A007716 counts non-isomorphic multiset partitions, connected A007718.
A129271 counts connected choosable simple graphs, unlabeled A005703.
A133686 counts choosable labeled graphs, covering A367869.
Cf. A000088, A000612, A006649, A014068, A055621, A137916, A137917, A368835, A369194, A369199.

Programs

  • Mathematica
    brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{(Union@@m)[[i]],p[[i]]},{i,Length[p]}])], {p,Permutations[Range[Length[Union@@m]]]}]]];
Table[Length[Union[brute /@ Select[Subsets[Subsets[Range[n],{1,2}]], Union@@#==Range[n]&&Length[Select[Tuples[#], UnsameQ@@#&]]!=0&]]],{n,0,4}]

Formula

First differences of A369145.
Euler transform of A369289 with A369289(1) = 1. - Andrew Howroyd, Feb 02 2024

Extensions

a(7) onwards from Andrew Howroyd, Feb 02 2024

A368834 Number of unlabeled simple graphs covering n vertices such that it is possible to choose a different vertex from each edge (choosable).

Original entry on oeis.org

1, 0, 1, 2, 5, 10, 27, 62, 165, 423, 1140, 3060, 8427, 23218, 64782, 181370, 511004, 1444285, 4097996, 11656644, 33243265, 94992847, 271953126, 779790166, 2239187466, 6438039076, 18532004323, 53400606823, 154024168401, 444646510812, 1284682242777
Offset: 0

Views

Author

Gus Wiseman, Jan 23 2024

Keywords

Examples

			Representatives of the a(2) = 1 through a(5) = 10 simple graphs:
  {12}  {12}{13}      {12}{34}          {12}{13}{45}
        {12}{13}{23}  {12}{13}{14}      {12}{13}{14}{15}
                      {12}{13}{24}      {12}{13}{14}{25}
                      {12}{13}{14}{23}  {12}{13}{23}{45}
                      {12}{13}{24}{34}  {12}{13}{24}{35}
                                        {12}{13}{14}{15}{23}
                                        {12}{13}{14}{23}{25}
                                        {12}{13}{14}{23}{45}
                                        {12}{13}{14}{25}{35}
                                        {12}{13}{24}{35}{45}

Links

Crossrefs

Without the choice condition we have A002494, labeled A006129.
The connected case is A005703, labeled A129271.
This is the covering case of A134964, complement A140637.
The labeled version is A367869, complement A367868.
The version with loops is A369200, complement A369147.
The complement is counted by A369202.
A007716 counts unlabeled multiset partitions, connected A007718.
A054548 counts graphs covering n vertices with k edges, with loops A369199.
A283877 counts unlabeled set-systems, connected A300913.
Cf. A000088, A006649, A001434, A055621, A116508, A133686, A137916, A137917, A368984, A369146.

Programs

  • Mathematica
    brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{(Union@@m)[[i]],p[[i]]},{i,Length[p]}])], {p,Permutations[Range[Length[Union@@m]]]}]]];
Table[Length[Union[brute /@ Select[Subsets[Subsets[Range[n],{2}]],Union@@#==Range[n] && Length[Select[Tuples[#],UnsameQ@@#&]]!=0&]]],{n,0,5}]

Formula

Euler transform of A005703 with A005703(1) = 0.
First differences of A134964.
a(n) = A002494(n) - A369202(n).

A368926 Triangle read by rows where T(n,k) is the number of unlabeled loop-graphs on n vertices with k loops and n-k non-loops such that it is possible to choose a different element from each edge.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 1, 2, 1, 1, 2, 5, 3, 1, 1, 5, 12, 7, 3, 1, 1, 14, 29, 19, 8, 3, 1, 1, 35, 75, 47, 21, 8, 3, 1, 1, 97, 191, 127, 54, 22, 8, 3, 1, 1, 264, 504, 331, 149, 56, 22, 8, 3, 1, 1, 733, 1339, 895, 395, 156, 57, 22, 8, 3, 1, 1
Offset: 0

Views

Author

Gus Wiseman, Jan 13 2024

Keywords

Comments

Also the number of unlabeled loop-graphs covering n vertices with k loops and n-k non-loops such that each connected component has the same number of edges as vertices.

Examples

			Triangle begins:
   1
   0  1
   0  1  1
   1  2  1  1
   2  5  3  1  1
   5 12  7  3  1  1
  14 29 19  8  3  1  1
  35 75 47 21  8  3  1  1

Links

Crossrefs

The case of a unique choice is A106234, row sums A000081.
Column k = 0 is A137917, labeled version A137916.
Without the choice condition we have A368836.
The labeled version is A368924, row sums maybe A333331.
Row sums are A368984, complement A368835.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A006125 counts graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
A014068 counts loop-graphs, unlabeled A000666.
A322661 counts labeled covering half-loop-graphs, connected A062740.
Cf. A057500, A116508, A133686, A367863, A367869, A368596, A368597, A368598, A368601, A368836, A368927.

Programs

  • Mathematica
    Table[Length[Union[sysnorm /@ Select[Subsets[Subsets[Range[n],{1,2}],{n}],Count[#,{_}]==k && Length[Select[Tuples[#],UnsameQ@@#&]]!=0&]]], {n,0,5},{k,0,n}]
  • PARI
    \\ TreeGf gives gf of A000081; G(n,1) is gf of A368983.
TreeGf(N)={my(A=vector(N, j, 1)); for (n=1, N-1, A[n+1] = 1/n * sum(k=1, n, sumdiv(k, d, d*A[d]) * A[n-k+1] ) ); x*Ser(A)}
G(n,y)={my(t=TreeGf(n)); my(g(e)=subst(t + O(x*x^(n\e)), x, x^e) + O(x*x^n)); 1 + (sum(d=1, n, eulerphi(d)/d*log(1/(1-g(d)))) + ((1+g(1))^2/(1-g(2))-1)/2 - (g(1)^2 + g(2)))/2 + (y-1)*g(1)}
EulerMTS(p)={my(n=serprec(p,x)-1,vars=variables(p)); exp(sum(i=1, n, substvec(p + O(x*x^(n\i)), vars, apply(v->v^i,vars))/i))}
T(n)={[Vecrev(p) | p <- Vec(EulerMTS(G(n,y) - 1))]}
{ my(A=T(8)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Jan 14 2024

Extensions

a(36) onwards from Andrew Howroyd, Jan 14 2024
Previous Showing 11-15 of 15 results.

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