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.

Showing 1-4 of 4 results.

A001434 Number of graphs with n nodes and n edges.

Original entry on oeis.org

1, 0, 0, 1, 2, 6, 21, 65, 221, 771, 2769, 10250, 39243, 154658, 628635, 2632420, 11353457, 50411413, 230341716, 1082481189, 5228952960, 25945377057, 132140242356, 690238318754, 3694876952577, 20252697246580, 113578669178222, 651178533855913, 3813856010041981
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

The labeled version is A116508. - Gus Wiseman, Feb 22 2024

Examples

			From _Gus Wiseman_, Feb 22 2024: (Start)
Representatives of the a(0) = 1 through a(5) = 6 graphs:
  {}  .  .  {12,13,23}  {12,13,14,23}  {12,13,14,15,23}
                        {12,13,24,34}  {12,13,14,23,24}
                                       {12,13,14,23,25}
                                       {12,13,14,23,45}
                                       {12,13,14,25,35}
                                       {12,13,24,35,45}
(End)

References

  • J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 146.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

The connected case is A001429, labeled A057500.
The covering case is A006649, labeled A367863.
Diagonal n = k of A008406.
The labeled version is A116508.
The version with loops is A368598, connected A368983.
Allowing up to n edges gives A370315, labeled A369192.
A000088 counts unlabeled graphs, labeled A006125.
A001349 counts unlabeled connected graphs, labeled A001187.
A002494 counts unlabeled covering graphs, labeled A006129.
Cf. A000055, A005703, A014068, A054924, A137916, A137917, A236570, A369201.

Programs

  • Mathematica
    (* first do *) Needs["Combinatorica`"] (* then *) Table[ NumberOfGraphs[n, n], {n, 24}] (* Robert G. Wilson v, Mar 22 2011 *)
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 /@ Subsets[Subsets[Range[n],{2}],{n}]]],{n,0,5}] (* Gus Wiseman, Feb 22 2024 *)
  • PARI
    a(n) = polcoef(G(n, O(x*x^n)), n) \\ G defined in A008406. - Andrew Howroyd, Feb 02 2024

Extensions

More terms from Vladeta Jovovic, Jan 07 2000
a(0)=1 prepended by Andrew Howroyd, Feb 02 2024

A370169 Number of unlabeled loop-graphs covering n vertices with at most n edges.

Original entry on oeis.org

1, 1, 3, 7, 19, 48, 135, 373, 1085, 3184, 9590, 29258, 90833, 285352, 908006, 2919953, 9487330, 31111997, 102934602, 343389708, 1154684849, 3912345408, 13353796977, 45906197103, 158915480378, 553897148543, 1943627750652, 6865605601382, 24411508473314, 87364180212671, 314682145679491
Offset: 0

Views

Author

Gus Wiseman, Feb 16 2024

Keywords

Examples

			The a(0) = 1 through a(4) = 19 loop-graph edge sets (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},{1,2},{3,4}}
                                               {{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

The case of equality is A368599, covering case of A368598.
The labeled version is A369194, covering case of A066383.
This is the covering case of A370168.
The loopless version is the covering case of A370315, labeled A369192.
This is the loopless version is A370316, labeled A369191.
A006125 counts graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
A322661 counts covering loop-graphs, unlabeled A322700.
Cf. A000666, A005703, A014068, A062740, A116508, A368597, A368927, A369141, A369196, A369197, 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[#]<=n&]]],{n,0,5}]
  • PARI
    \\ G defined in A070166.
a(n)=my(A=O(x*x^n)); if(n==0, 1, polcoef((G(n,A)-G(n-1,A))/(1-x), n)) \\ Andrew Howroyd, Feb 19 2024

Extensions

a(7) onwards from Andrew Howroyd, Feb 19 2024

A370168 Number of unlabeled loop-graphs with n vertices and at most n edges.

Original entry on oeis.org

1, 2, 5, 13, 36, 102, 313, 994, 3318, 11536, 41748, 156735, 609973, 2456235, 10224216, 43946245, 194866898, 890575047, 4190997666, 20289434813, 100952490046, 515758568587, 2703023502100, 14518677321040, 79852871813827, 449333028779385, 2584677513933282
Offset: 0

Views

Author

Gus Wiseman, Feb 16 2024

Keywords

Examples

			The a(0) = 1 through a(3) = 13 loop-graph edge sets (loops shown as singletons):
  {}  {}     {}           {}
      {{1}}  {{1}}        {{1}}
             {{1,2}}      {{1,2}}
             {{1},{2}}    {{1},{2}}
             {{1},{1,2}}  {{1},{1,2}}
                          {{1},{2,3}}
                          {{1,2},{1,3}}
                          {{1},{2},{3}}
                          {{1},{2},{1,2}}
                          {{1},{2},{1,3}}
                          {{1},{1,2},{1,3}}
                          {{1},{1,2},{2,3}}
                          {{1,2},{1,3},{2,3}}

Links

Crossrefs

The labeled version is A066383, covering A369194.
The case of equality is A368598, covering A368599.
The covering case is A370169, labeled A369194.
The loopless version is A370315, labeled A369192.
The covering loopless version is A370316, labeled A369191.
A006125 counts graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
A322661 counts covering loop-graphs, unlabeled A322700.
Cf. A000666, A005703, A014068, A062740, A368597, A368927, A369141, A369196, A369197, 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}]],Length[#]<=n&]]],{n,0,5}]
  • PARI
    a(n)=my(A=O(x*x^n)); if(n==0, 1, polcoef(G(n, A)/(1-x), n)) \\ G defined in A070166. - Andrew Howroyd, Feb 19 2024

Extensions

a(7) onwards from Andrew Howroyd, Feb 19 2024

A370316 Number of unlabeled simple graphs covering n vertices with at most n edges.

Original entry on oeis.org

1, 0, 1, 2, 5, 10, 28, 68, 193, 534, 1568, 4635, 14146, 43610, 137015, 435227, 1400058, 4547768, 14917504, 49348612, 164596939, 553177992, 1872805144, 6385039022, 21917878860, 75739158828, 263438869515, 922219844982, 3249042441125, 11519128834499, 41097058489426
Offset: 0

Views

Author

Gus Wiseman, Feb 18 2024

Keywords

Examples

			The a(0) = 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

The connected case is A005703, labeled A129271.
The case of exactly n edges is A006649, covering case of A001434.
The labeled version is A369191.
Partial row sums of A370167, covering case of A008406.
The non-covering version with loops is A370168, labeled A066383.
The version with loops is A370169, labeled A369194.
The non-covering version is A370315, labeled A369192.
A006125 counts graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
A322661 counts covering loop-graphs, unlabeled A322700.
Cf. A054548, A062734, A116508, A367862, A367863, A368599, A369193.

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}],{0,n}], Union@@#==Range[n]&]]],{n,0,5}]
  • PARI
    \\ G defined in A008406.
a(n)=my(A=O(x*x^n)); if(n==0, 1, polcoef((G(n,A)-G(n-1,A))/(1-x), n)) \\ Andrew Howroyd, Feb 19 2024

Extensions

a(8) onwards from Andrew Howroyd, Feb 19 2024
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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