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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A368731 Number of non-isomorphic n-element sets of nonempty subsets of {1..n}.

Original entry on oeis.org

1, 1, 2, 10, 97, 2160, 126862, 21485262, 11105374322, 18109358131513, 95465831661532570, 1660400673336788987026, 96929369602251313489896310, 19268528295096123543660356281600, 13203875101002459910158494602665950757, 31517691852305548841992346407978317698725021
Offset: 0

Views

Author

Gus Wiseman, Jan 07 2024

Keywords

Examples

			Non-isomorphic representatives of the a(3) = 10 set-systems:
  {{1},{2},{3}}
  {{1},{2},{1,2}}
  {{1},{2},{1,3}}
  {{1},{2},{1,2,3}}
  {{1},{1,2},{1,3}}
  {{1},{1,2},{2,3}}
  {{1},{1,2},{1,2,3}}
  {{1},{2,3},{1,2,3}}
  {{1,2},{1,3},{2,3}}
  {{1,2},{1,3},{1,2,3}}

Links

Crossrefs

The case of graphs is A001434, labeled A116508.
Labeled version is A136556, covering A054780, binomial transform of A367916.
The case of labeled covering graphs is A367863, binomial transform A367862.
These include the set-systems ranked by A367917.
The covering case is A368186, for graphs A006649, connected A057500.
Requiring all edges to be singletons or pairs gives A368598.
A003465 counts covers with any number of edges, unlabeled A055621.
A046165 counts minimal covers, ranks A309326.
A058891 counts set-systems, unlabeled A000612, without singletons A016031.
A059201 counts covering T_0 set-systems, unlabeled A319637, ranks A326947.
Cf. A000088, A007716, A283877, A367901, A367902, A367903, A368599.

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 /@ Subsets[Subsets[Range[n],{1,n}],{n}]]],{n,0,4}]
  • PARI
    a(n) = polcoef(G(n, n), n) \\ G defined in A368186. - Andrew Howroyd, Jan 11 2024

Extensions

Terms a(6) and beyond from Andrew Howroyd, Jan 11 2024

A369202 Number of unlabeled simple 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, 0, 0, 2, 13, 95, 826, 11137, 261899, 11729360, 1006989636, 164072166301, 50336940172142, 29003653625802754, 31397431814146891910, 63969589218557753075156, 245871863137828405124380563, 1787331789281458167615190373076, 24636021675399858912682459601585276
Offset: 0

Views

Author

Gus Wiseman, Jan 23 2024

Keywords

Comments

These are simple graphs covering n vertices such that some connected component has at least two cycles.

Examples

			Representatives of the a(4) = 2 and a(5) = 13 simple graphs:
  {12}{13}{14}{23}{24}      {12}{13}{14}{15}{23}{24}
  {12}{13}{14}{23}{24}{34}  {12}{13}{14}{15}{23}{45}
                            {12}{13}{14}{23}{24}{35}
                            {12}{13}{14}{23}{25}{45}
                            {12}{13}{14}{25}{35}{45}
                            {12}{13}{14}{15}{23}{24}{25}
                            {12}{13}{14}{15}{23}{24}{34}
                            {12}{13}{14}{15}{23}{24}{35}
                            {12}{13}{14}{23}{24}{35}{45}
                            {12}{13}{14}{15}{23}{24}{25}{34}
                            {12}{13}{14}{15}{23}{24}{35}{45}
                            {12}{13}{14}{15}{23}{24}{25}{34}{35}
                            {12}{13}{14}{15}{23}{24}{25}{34}{35}{45}

Links

Crossrefs

Without the choice condition we have A002494, labeled A006129.
The connected case is A140636.
This is the covering case of A140637, complement A134964.
The labeled version is A367868, complement A367869.
The complement is counted by A368834.
The version with loops is A369147, complement A369200.
A005703 counts unlabeled connected choosable simple graphs, labeled A129271.
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, A000612, A006649, A001434, A055621, A137916, A137917, A140638, A368596, A369141, 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

First differences of A140637.
a(n) = A002494(n) - A368834(n).

A370315 Number of unlabeled simple graphs with n possibly isolated vertices and up to n edges.

Original entry on oeis.org

1, 1, 2, 4, 9, 20, 54, 146, 436, 1372, 4577, 15971, 58376, 221876, 876012, 3583099, 15159817, 66248609, 298678064, 1387677971, 6637246978, 32648574416, 165002122350, 855937433641, 4553114299140, 24813471826280, 138417885372373, 789683693019999, 4603838061688077
Offset: 0

Views

Author

Gus Wiseman, Feb 18 2024

Keywords

Examples

			The a(1) = 1 through a(4) = 9 graph edge sets:
  {}  {}    {}          {}
      {12}  {12}        {12}
            {12-13}     {12-13}
            {12-13-23}  {12-34}
                        {12-13-14}
                        {12-13-23}
                        {12-13-24}
                        {12-13-14-23}
                        {12-13-24-34}

Links

Crossrefs

The case of exactly n edges is A001434, covering A006649.
The connected covering case is A005703, labeled A129271.
Partial row sums of A008406, covering A370167.
The labeled version is A369192.
The version with loops is A370168, labeled A066383.
The covering case is A370316, labeled A369191.
A006125 counts graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
Cf. A000666, A322661, A322700, A369194, A369196, A369197, A369199, A370169.

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}]], Length[#]<=n&]]],{n,0,5}]
  • PARI
    a(n) = if(n<=1, n>=0, polcoef(G(n, O(x*x^n))/(1-x),n)) \\ G(n) defined in A008406. - Andrew Howroyd, Feb 20 2024

Formula

Sum of first n+1 terms of row n of A008406.

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).

A006648 Number of graphs with n nodes, n-1 edges and no isolated vertices.

Original entry on oeis.org

1, 1, 2, 4, 9, 20, 50, 124, 332, 895, 2513, 7172, 20994, 62366, 188696, 578717, 1799999, 5666257, 18047319, 58097540, 188953756, 620493315, 2056582095, 6877206111, 23195975865, 78891742748, 270505303760, 934890953041, 3256230606767
Offset: 2

Views

Author

N. J. A. Sloane

Keywords

References

  • W. L. Kocay, Some new methods in reconstruction theory, pp. 89 - 114 of Combinatorial Mathematics IX. Proc. Ninth Australian Conference (Brisbane, August 1981). Ed. E. J. Billington, S. Oates-Williams and A. P. Street. Lecture Notes Math., 952. Springer-Verlag, 1982.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Crossrefs

Cf. A001433, A001434.

Formula

a(n) = A001433(n) - A001434(n - 1). - Sean A. Irvine, Jun 06 2017

Extensions

More terms from Vladeta Jovovic, Mar 02 2008
More terms from Sean A. Irvine, Jun 06 2017
Previous Showing 11-15 of 15 results.

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

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