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 11-18 of 18 results.

A369196 Number of labeled loop-graphs with n vertices and at most as many edges as covered vertices.

Original entry on oeis.org

1, 2, 7, 39, 320, 3584, 51405, 900947, 18661186, 445827942, 12062839691, 364451604095, 12157649050827, 443713171974080, 17583351295466338, 751745326170662049, 34485624653535808340, 1689485711682987916502, 88030098291829749593643, 4860631073631586486397141
Offset: 0

Views

Author

Gus Wiseman, Jan 17 2024

Keywords

Examples

			The a(0) = 1 through a(2) = 7 loop-graphs:
  {}  {}     {}
      {{1}}  {{1}}
             {{2}}
             {{1,2}}
             {{1},{2}}
             {{1},{1,2}}
             {{2},{1,2}}

Crossrefs

The version counting all vertices is A066383, without loops A369192.
The loopless case is A369193, with case of equality A367862.
The covering case is A369194, connected A369197, minimal case A001862.
The case of equality is A369198, covering case A368597.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A006125 counts simple graphs, also loop-graphs if shifted left.
A006129 counts covering graphs, unlabeled A002494.
A054548 counts graphs covering n vertices with k edges, with loops A369199.
A322661 counts covering loop-graphs, unlabeled A322700.
A368927 counts choosable loop-graphs, covering A369140.
A369141 counts non-choosable loop-graphs, covering A369142.
Cf. A000169, A000272, A000666, A001187, A006649, A057500, A062740, A116508, A367863, A369145.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Subsets[Range[n],{1,2}]],Length[#]<=Length[Union@@#]&]],{n,0,5}]

Formula

Binomial transform of A369194.

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

A370317 Number of labeled graphs with n vertices (allowing isolated vertices) and n edges, such that the edge set is connected.

Original entry on oeis.org

1, 0, 0, 1, 15, 252, 4905, 110715, 2864148, 83838720, 2744568522, 99463408335, 3955626143040, 171344363805582, 8031863998136355, 405150528051451000, 21884686370917378050, 1260420510502767861840, 77105349570138633021624, 4993117552678619556356085
Offset: 0

Views

Author

Gus Wiseman, Feb 17 2024

Keywords

Examples

			The a(0) = 0 through a(4) = 15 graphs:
  {}  .  .  {{1,2},{1,3},{2,3}}  {{1,2},{1,3},{1,4},{2,3}}
                                 {{1,2},{1,3},{1,4},{2,4}}
                                 {{1,2},{1,3},{1,4},{3,4}}
                                 {{1,2},{1,3},{2,3},{2,4}}
                                 {{1,2},{1,3},{2,3},{3,4}}
                                 {{1,2},{1,3},{2,4},{3,4}}
                                 {{1,2},{1,4},{2,3},{2,4}}
                                 {{1,2},{1,4},{2,3},{3,4}}
                                 {{1,2},{1,4},{2,4},{3,4}}
                                 {{1,2},{2,3},{2,4},{3,4}}
                                 {{1,3},{1,4},{2,3},{2,4}}
                                 {{1,3},{1,4},{2,3},{3,4}}
                                 {{1,3},{1,4},{2,4},{3,4}}
                                 {{1,3},{2,3},{2,4},{3,4}}
                                 {{1,4},{2,3},{2,4},{3,4}}

Links

Crossrefs

The covering case is A057500.
This is the connected case of A116508.
Allowing any number of edges gives A287689.
Counting only covered vertices gives A370318.
A006125 counts graphs, unlabeled A000088.
A006129 counts covering graphs, connected A001187.
A369192 counts graphs with at most n edges, covering A369191.
Cf. A000169, A001862, A054780, A062740, A129271, A367863, A369193, A369197.

Programs

  • Mathematica
    csm[s_]:=With[{c=Select[Subsets[Range[Length[s]], {2}],Length[Intersection@@s[[#]]]>0&]},If[c=={},s, csm[Sort[Append[Delete[s,List/@c[[1]]], Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[Subsets[Subsets[Range[n], {2}]],Length[#]==n&&Length[csm[#]]<=1&]], {n,0,5}]
  • PARI
    a(n)=n!*polcoef(polcoef(exp(x + O(x*x^n))*(1 + log(sum(k=0, n, (1 + y + O(y*y^n))^binomial(k,2)*x^k/k!, O(x*x^n)))), n), n) \\ Andrew Howroyd, Feb 19 2024

Formula

a(n) = n!*[x^n][y^n] exp(x)*(1 + log(Sum_{k>=0} (1 + y)^binomial(k, 2)*x^k/k!)). - Andrew Howroyd, Feb 19 2024

Extensions

a(8) 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

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.

A370318 Number of labeled simple graphs with n vertices and the same number of edges as covered vertices, such that the edge set is connected.

Original entry on oeis.org

0, 0, 0, 1, 19, 307, 5237, 99137, 2098946, 49504458, 1291570014, 37002273654, 1156078150969, 39147186978685, 1428799530304243, 55933568895261791, 2338378885159906196, 103995520598384132516, 4903038902046860966220, 244294315694676224001852, 12827355456239840407125363
Offset: 0

Views

Author

Gus Wiseman, Feb 18 2024

Keywords

Comments

The case of an empty edge set is excluded.

Links

Crossrefs

The covering case is A057500, which is also the covering case of A370317.
This is the connected case of A367862, covering A367863.
A001187 counts connected graphs, A001349 unlabeled.
A006125 counts graphs, A000088 unlabeled.
A006129 counts covering graphs, A002494 unlabeled.
A062734 counts connected graphs by edge count.
A133686 = graphs satisfy strict AoC, connected A129271, covering A367869.
A143543 counts simple labeled graphs by number of connected components.
A367867 = graphs contradict strict AoC, connected A140638, covering A367868.
Cf. A001429, A006649, A061540, A116508, A323818, A367916, A368951, A369197.

Programs

  • Mathematica
    csm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}], Length[Intersection@@s[[#]]]>0&]},If[c=={},s, csm[Sort[Append[Delete[s,List/@c[[1]]], Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[Subsets[Subsets[Range[n], {2}]],Length[#]==Length[Union@@#] && Length[csm[#]]==1&]],{n,0,5}]
  • PARI
    \\ Compare A370317; use A057500 for efficiency.
a(n)=n!*polcoef(polcoef(exp(x*y + O(x*x^n))*(-x+log(sum(k=0, n, (1 + y + O(y*y^n))^binomial(k, 2)*x^k/k!, O(x*x^n)))), n), n) \\ Andrew Howroyd, Feb 19 2024

Formula

Binomial transform of A057500 (if the null graph is not connected).
a(n) = n!*[x^n][y^n] exp(x*y)*(-x + log(Sum_{k>=0} (1 + y)^binomial(k, 2)*x^k/k!)). - Andrew Howroyd, Feb 19 2024

A369195 Irregular triangle read by rows where T(n,k) is the number of labeled connected loop-graphs covering n vertices with k edges.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 1, 0, 0, 3, 10, 12, 6, 1, 0, 0, 0, 16, 79, 162, 179, 116, 45, 10, 1, 0, 0, 0, 0, 125, 847, 2565, 4615, 5540, 4720, 2948, 1360, 455, 105, 15, 1, 0, 0, 0, 0, 0, 1296, 11436, 47100, 121185, 220075, 301818, 325578, 282835, 200115, 115560, 54168, 20343, 5985, 1330, 210, 21, 1
Offset: 0

Views

Author

Gus Wiseman, Jan 19 2024

Keywords

Comments

This sequence excludes the graph consisting of a single isolated vertex without a loop. - Andrew Howroyd, Feb 02 2024

Examples

			Triangle begins:
    1
    0    1
    0    1    2    1
    0    0    3   10   12    6    1
    0    0    0   16   79  162  179  116   45   10    1
Row n = 3 counts the following loop-graphs (loops shown as singletons):
  .  .  {12,13}  {1,12,13}   {1,2,12,13}   {1,2,3,12,13}   {1,2,3,12,13,23}
        {12,23}  {1,12,23}   {1,2,12,23}   {1,2,3,12,23}
        {13,23}  {1,13,23}   {1,2,13,23}   {1,2,3,13,23}
                 {2,12,13}   {1,3,12,13}   {1,2,12,13,23}
                 {2,12,23}   {1,3,12,23}   {1,3,12,13,23}
                 {2,13,23}   {1,3,13,23}   {2,3,12,13,23}
                 {3,12,13}   {1,12,13,23}
                 {3,12,23}   {2,3,12,13}
                 {3,13,23}   {2,3,12,23}
                 {12,13,23}  {2,3,13,23}
                             {2,12,13,23}
                             {3,12,13,23}

Links

Crossrefs

Row lengths are A000124.
Diagonal T(n,n-1) is A000272, rooted A000169.
The case without loops is A062734.
Row sums are A062740.
Transpose is A322147.
Column sums are A322151.
Diagonal T(n,n) is A368951, connected case of A368597.
Connected case of A369199, without loops A054548.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A000666 counts unlabeled loop-graphs.
A001187 counts connected graphs, unlabeled A001349.
A006125 counts simple graphs, also loop-graphs if shifted left.
A006129 counts covering graphs, unlabeled A002494.
A322661 counts covering loop-graphs, unlabeled A322700.
A368927 counts choosable loop-graphs, covering A369140.
A369141 counts non-choosable loop-graphs, covering A369142.
Cf. A001862, A014068, A054923, A057500, A066383, A322114, A322137, A369197.

Programs

  • Mathematica
    csm[s_]:=With[{c=Select[Subsets[Range[Length[s]], {2}],Length[Intersection@@s[[#]]]>0&]},If[c=={},s, csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[Subsets[Subsets[Range[n],{1,2}],{k}], Length[Union@@#]==n&&Length[csm[#]]<=1&]], {n,0,5},{k,0,Binomial[n+1,2]}]
  • PARI
    T(n)={[Vecrev(p) | p<-Vec(serlaplace(1 - x + log(sum(j=0, n, (1 + y)^binomial(j+1, 2)*x^j/j!, O(x*x^n))))) ]}
{ my(A=T(6)); for(i=1, #A, print(A[i])) } \\ Andrew Howroyd, Feb 02 2024

Formula

E.g.f.: 1 - x + log(Sum_{j >= 0} (1 + y)^binomial(j+1, 2)*x^j/j!). - Andrew Howroyd, Feb 02 2024

A369198 Number of labeled loop-graphs with n vertices and the same number of edges as covered vertices.

Original entry on oeis.org

1, 2, 6, 30, 241, 2759, 40824, 736342, 15622835, 380668095, 10467815086, 320529284621, 10813165015074, 398413594789777, 15917197015926392, 685312404706694574, 31631317971844128229, 1558017329350990780607, 81567807853701988869120, 4522975947689168088308305
Offset: 0

Views

Author

Gus Wiseman, Jan 18 2024

Keywords

Examples

			The a(0) = 1 through a(3) = 30 loop-graphs (loops shown as singletons):
  {}  {}     {}           {}
      {{1}}  {{1}}        {{1}}
             {{2}}        {{2}}
             {{1},{2}}    {{3}}
             {{1},{1,2}}  {{1},{2}}
             {{2},{1,2}}  {{1},{3}}
                          {{2},{3}}
                          {{1},{1,2}}
                          {{1},{1,3}}
                          {{2},{1,2}}
                          {{2},{2,3}}
                          {{3},{1,3}}
                          {{3},{2,3}}
                          {{1},{2},{3}}
                          {{1},{2},{1,3}}
                          {{1},{2},{2,3}}
                          {{1},{3},{1,2}}
                          {{1},{3},{2,3}}
                          {{2},{3},{1,2}}
                          {{2},{3},{1,3}}
                          {{1},{1,2},{1,3}}
                          {{1},{1,2},{2,3}}
                          {{1},{1,3},{2,3}}
                          {{2},{1,2},{1,3}}
                          {{2},{1,2},{2,3}}
                          {{2},{1,3},{2,3}}
                          {{3},{1,2},{1,3}}
                          {{3},{1,2},{2,3}}
                          {{3},{1,3},{2,3}}
                          {{1,2},{1,3},{2,3}}

Crossrefs

The version counting all vertices is A014068.
The loopless case is A367862, counting all vertices A116508.
The covering case is A368597, connected A368951.
With inequality we have A369196, covering A369194, connected A369197.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A006125 counts simple graphs, also loop-graphs if shifted left.
A006129 counts covering graphs, unlabeled A002494.
A054548 counts graphs covering n vertices with k edges, with loops A369199.
A322661 counts covering loop-graphs, unlabeled A322700.
A368927 counts choosable loop-graphs, covering A369140.
A369141 counts non-choosable loop-graphs, covering A369142.
Cf. A000272, A000666, A006649, A062740, A066383, A367863, A369192, A369193.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Subsets[Range[n],{1,2}]], Length[#]==Length[Union@@#]&]],{n,0,5}]

Formula

Binomial transform of A368597.
Previous Showing 11-18 of 18 results.

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