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 1-3 of 3 results.

A005703 Number of n-node connected graphs with at most one cycle.

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 19, 44, 112, 287, 763, 2041, 5577, 15300, 42419, 118122, 330785, 929469, 2621272, 7411706, 21010378, 59682057, 169859257, 484234165, 1382567947, 3952860475, 11315775161, 32430737380, 93044797486, 267211342954, 768096496093, 2209772802169
Offset: 0

Views

Author

N. J. A. Sloane, R. K. Guy

Keywords

Comments

a(n) is the number of pseudotrees on n nodes. - Eric W. Weisstein, Jun 11 2012
Also unlabeled connected graphs covering n vertices with at most n edges. For this definition we have a(1) = 0 and possibly a(0) = 0. - Gus Wiseman, Feb 20 2024

Examples

			From _Gus Wiseman_, Feb 20 2024: (Start)
Representatives of the a(0) = 1 through a(5) = 8 graphs:
  {}  .  {12}  {12,13}     {12,13,14}     {12,13,14,15}
               {12,13,23}  {12,13,24}     {12,13,14,25}
                           {12,13,14,23}  {12,13,24,35}
                           {12,13,24,34}  {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}
(End)

References

  • J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 150.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A000055, A000081, A001429 (labeled A057500), A134964 (number of pseudoforests, labeled A133686).
The labeled version is A129271.
The connected complement is A140636, labeled A140638.
Non-connected: A368834 (labeled A367869) or A370316 (labeled A369191).
A001187 counts connected graphs, unlabeled A001349.
A006125 counts simple graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
A062734 counts connected graphs by number of edges.
Cf. A000272, A006649, A116508, A140637, A143543, A367862, A367863, A368951, A369197, A370317, A370318.

Programs

  • Mathematica
    Needs["Combinatorica`"]; nn = 20; t[x_] := Sum[a[n] x^n, {n, 1, nn}];
a[0] = 0;
b = Drop[Flatten[
    sol = SolveAlways[
      0 == Series[
        t[x] - x Product[1/(1 - x^i)^a[i], {i, 1, nn}], {x, 0, nn}],
      x]; Table[a[n], {n, 0, nn}] /. sol], 1];
r[x_] := Sum[b[[n]] x^n, {n, 1, nn}]; c =
Drop[Table[
    CoefficientList[
     Series[CycleIndex[DihedralGroup[n], s] /.
       Table[s[i] -> r[x^i], {i, 1, n}], {x, 0, nn}], x], {n, 3,
     nn}] // Total, 1];
d[x_] := Sum[c[[n]] x^n, {n, 1, nn}]; CoefficientList[
Series[r[x] - (r[x]^2 - r[x^2])/2 + d[x] + 1, {x, 0, nn}], x] (* Geoffrey Critzer, Nov 17 2014 *)
  • 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 + g(1) + (g(2) - g(1)^2)/2 + sum(k=3, n, sumdiv(k, d, eulerphi(d)*g(d)^(k/d))/k + if(k%2, g(1)*g(2)^(k\2), (g(1)^2+g(2))*g(2)^(k/2-1)/2))/2)}; \\ Andrew Howroyd and Washington Bomfim, May 15 2021

Formula

a(n) = A000055(n) + A001429(n).

Extensions

More terms from Vladeta Jovovic, Apr 19 2000 and from Michael Somos, Apr 26 2000
a(27) corrected and a(28) and a(29) computed by Washington Bomfim, May 14 2008

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

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).
Showing 1-3 of 3 results.

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

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