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.

Previous Showing 31-35 of 35 results.

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).

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

A210507 Number of labeled graphs on [n] with unicyclic components containing a given edge.

Original entry on oeis.org

1, 10, 111, 1468, 22940, 416250, 8626660, 201349672, 5230931454, 149783426470, 4688281021490, 159284662406460, 5838769123729984, 229711022253150382, 9655348958575618320, 431845990498159342000, 20479127764425617465660, 1026429489947790074019978
Offset: 3

Views

Author

Gary Gordon, Jan 25 2013

Keywords

Comments

This gives the number of matroid bases that contain a given element (edge) of the bicircular matroid of K_n.

Examples

			a(4)=10 means that 10 (of the 15) labeled unicyclic graphs on 4 vertices contain a given edge.

References

  • O. Giménez, A. de Mier, M. Noy, On the Number of Bases of Bicircular Matroids, Ann. Comb. 9 (2005), no. 1, 35-45.

Crossrefs

Cf. A137916.

Formula

a(n) = 2*b(n)/(n-1), where b(n) is seq A137916.

A217763 Triangular array read by rows: T(n,k) is the number of simple labeled graphs on n nodes with unicyclic components having exactly k nodes with degree 1; n>=3, 0<=k<=n-3.

Original entry on oeis.org

1, 3, 12, 12, 90, 120, 70, 600, 1800, 1200, 465, 4725, 19530, 31500, 12600, 3507, 42168, 211680, 529200, 529200, 141120, 30016, 414288, 2451456, 7902720, 13124160, 8890560, 1693440, 286884, 4460760, 30413880, 117573120, 266716800, 312439680, 152409600, 21772800
Offset: 3

Views

Author

Geoffrey Critzer, Mar 23 2013

Keywords

Comments

Column k=0 is A001205.
Row sums are A137916.

Examples

			  ....o-o..........o-o......
  ....| |..........|\ ......
  ....o-o..........o-o......
  T(4,0)=3 because the graph on the left has 4 nodes and 0 nodes with degree 1. It has 3 labelings.
  T(4,1)=12 because the graph on the right has 4 nodes and 1 node with degree 1.  It has 12 labelings.
1,
3,     12,
12,    90,     120,
70,    600,    1800,    1200,
465,   4725,   19530,   31500,   12600,
3507,  42168,  211680,  529200,  529200,   141120,
30016, 414288, 2451456, 7902720, 13124160, 8890560, 1693440.

Programs

  • Mathematica
    nn=10;f[list_]:=Select[list,#>0&];t=Sum[Sum[n!/k! StirlingS2[n-1,n-k]y^k x^n/n!,{k,1,n}],{n,0,nn}];Map[Reverse,Map[f,Drop[Range[0,nn]!CoefficientList[Series[ Exp[Log[1/(1-t)]/2-t/2-t^2/4],{x,0,nn}],{x,y}],3]]]//Grid

Formula

exp(A(B(x,y)), where A(x) is e.g.f. for A137916 and B(x,y) is e.g.f. for A055302, gives T(n,n-k) (offset).
Previous Showing 31-35 of 35 results.

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