A054548 -id:A054548 - cp's OEIS frontend

A369145 Number of unlabeled loop-graphs with up to n vertices such that it is possible to choose a different vertex from each edge (choosable).

1, 2, 5, 12, 30, 73, 185, 467, 1207, 3147, 8329, 22245, 60071, 163462, 448277, 1236913, 3432327, 9569352, 26792706, 75288346, 212249873, 600069431, 1700826842, 4831722294, 13754016792, 39224295915, 112048279650, 320563736148, 918388655873, 2634460759783, 7566000947867

Offset: 0

Gus Wiseman, Jan 22 2024

nonn

a(n) is the number of graphs with loops on n unlabeled vertices with every connected component having no more edges than vertices. - Andrew Howroyd, Feb 02 2024

			The a(0) = 1 through a(3) = 12 loop-graphs (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},{1,3}}
                          {{1},{1,2},{1,3}}
                          {{1},{1,2},{2,3}}
                          {{1,2},{1,3},{2,3}}

Andrew Howroyd, Table of n, a(n) for n = 0..500

Without the choice condition we get A000666, labeled A006125 (shifted left).
The case of a unique choice is A087803, labeled A088957.
Without loops we have A134964, labeled A133686 (covering A367869).
For exactly n edges and no loops we have A137917, labeled A137916.
The labeled version is A368927, covering A369140.
The labeled complement is A369141, covering A369142.
For exactly n edges we have A368984, labeled A333331 (maybe).
The complement for exactly n edges is A368835, labeled A368596.
The complement is counted by A369146, labeled A369141 (covering A369142).
The covering case is A369200.
The complement for exactly n edges and no loops is A369201, labeled A369143.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A006129 counts covering graphs, unlabeled A002494.
A054548 counts graphs covering n vertices with k edges, with loops A369199.
A129271 counts connected choosable simple graphs, unlabeled A005703.
A322661 counts labeled covering loop-graphs, unlabeled A322700.
A367867 counts non-choosable labeled graphs, covering A367868.
A368927 counts choosable labeled loop-graphs, covering A369140.
Cf. A000088, A006649, A062740, A140638, A368924, A369194.

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[Select[Tuples[#], UnsameQ@@#&]]!=0&]]],{n,0,4}]

Partial sums of A369200.

Euler transform of A369289. - Andrew Howroyd, Feb 02 2024

a(7) onwards from Andrew Howroyd, Feb 02 2024

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

Gus Wiseman, Jan 17 2024

nonn

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

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.

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

Binomial transform of A369194.

A369193 Number of labeled simple graphs with n vertices and at most as many edges as covered (non-isolated) vertices.

Original entry on oeis.org

1, 1, 2, 8, 57, 608, 8614, 151365, 3162353, 76359554, 2088663444, 63760182536, 2147325661180, 79051734050283, 3157246719905273, 135938652662043977, 6275929675565965599, 309242148569525451140, 16197470691388774460758, 898619766673014862321176, 52639402023471657682257626

Offset: 0

Gus Wiseman, Jan 17 2024

nonn

			The a(0) = 1 through a(3) = 8 graphs:
  {}  {}  {}       {}
          {{1,2}}  {{1,2}}
                   {{1,3}}
                   {{2,3}}
                   {{1,2},{1,3}}
                   {{1,2},{2,3}}
                   {{1,3},{2,3}}
                   {{1,2},{1,3},{2,3}}

The case of equality is A367862, covering case of A116508, also A367863.
The covering case is A369191, for loop-graphs A369194.
The version counting all vertices is A369192.
The version for loop-graphs is A369196, counting all vertices A066383.
A006125 counts simple graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
A054548 counts graphs covering n vertices with k edges, with loops A369199.
A133686 counts choosable graphs, covering A367869.
A367867 counts non-choosable graphs, covering A367868.
Cf. A000169, A000272, A000666, A001187, A006649, A053530, A129271, A143543, A322661, A367916.

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

Binomial transform of A369191.

A001862 Number of forests of least height with n nodes.

Original entry on oeis.org

1, 1, 2, 7, 26, 111, 562, 3151, 19252, 128449, 925226, 7125009, 58399156, 507222535, 4647051970, 44747776651, 451520086208, 4761032807937, 52332895618066, 598351410294193, 7102331902602676, 87365859333294151, 1111941946738682522, 14621347433458883187

Offset: 0

N. J. A. Sloane

nonn

From Gus Wiseman, Feb 14 2024: (Start)
Also the number of minimal loop-graphs covering n vertices. This is the minimal case of A322661. For example, the a(0) = 1 through a(3) = 7 loop-graphs are (loops represented as singletons):
  {}  {1}  {12}   {1-23}
           {1-2}  {2-13}
                  {3-12}
                  {12-13}
                  {12-23}
                  {13-23}
                  {1-2-3}
(End)

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983. See (3.3.7): number of ways to cover the complete graph K_n with star graphs.
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).

T. D. Noe, Table of n, a(n) for n = 0..100
John Riordan, Forests of labeled trees, J. Combin. Theory, 5 (1968), 90-103.
John Riordan, Letter to N. J. A. Sloane, Oct. 1970
John Riordan and N. J. A. Sloane, Correspondence, 1974

The connected case is A000272.
Without loops we have A053530, minimal case of A369191.
This is the minimal case of A322661.
A000666 counts unlabeled loop-graphs, covering A322700.
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.
Cf. A000085, A000169, A003465, A062740, A066383, A133686, A368597.

Range[0, 20]! CoefficientList[Series[Exp[x Exp[x] - x^2/2], {x, 0, 20}], x] (* Geoffrey Critzer, Mar 13 2011 *)
fasmin[y_]:=Complement[y,Union@@Table[Union[s,#]& /@ Rest[Subsets[Complement[Union@@y,s]]],{s,y}]];
Table[Length@fasmin[Select[Subsets[Subsets[Range[n],{1,2}]], Union@@#==Range[n]&]],{n,0,4}] (* Gus Wiseman, Feb 14 2024 *)

E.g.f.: exp(x*(exp(x)-x/2)).

Binomial transform of A053530. - Gus Wiseman, Feb 14 2024

Formula and more terms from Vladeta Jovovic, Mar 27 2001

A369201 Number of unlabeled simple graphs with n vertices and n edges 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, 0, 1, 7, 30, 124, 507, 2036, 8216, 33515, 138557, 583040, 2503093, 10985364, 49361893, 227342301, 1073896332, 5204340846, 25874724616, 131937166616, 689653979583, 3693193801069, 20247844510508, 113564665880028, 651138092719098, 3813739129140469

Offset: 0

Gus Wiseman, Jan 22 2024

nonn

These are graphs with n vertices and n edges having at least two cycles in the same component.

			The a(0) = 0 through a(6) = 7 simple graphs:
  .  .  .  .  .  {{12}{13}{14}{23}{24}}  {{12}{13}{14}{15}{23}{24}}
                                         {{12}{13}{14}{15}{23}{45}}
                                         {{12}{13}{14}{23}{24}{34}}
                                         {{12}{13}{14}{23}{24}{35}}
                                         {{12}{13}{14}{23}{24}{56}}
                                         {{12}{13}{14}{23}{25}{45}}
                                         {{12}{13}{14}{25}{35}{45}}

Andrew Howroyd, Table of n, a(n) for n = 0..50
Gus Wiseman, The a(6) = 7 unlabeled non-choosable graphs with 6 vertices and 6 edges.

Without the choice condition we have A001434, covering A006649.
The labeled version without choice is A116508, covering A367863, A367862.
The complement is counted by A137917, labeled A137916.
For any number of edges we have A140637, complement A134964.
For labeled set-systems we have A368600.
The case with loops is A368835, labeled A368596.
The labeled version is A369143, covering A369144.
A006129 counts covering graphs, unlabeled A002494.
A007716 counts unlabeled multiset partitions, connected A007718.
A054548 counts graphs covering n vertices with k edges, with loops A369199.
A129271 counts connected choosable simple graphs, unlabeled A005703.
Cf. A000088, A000612, A014068, A053530, A133686, A140638, A368601, A369141, A369146.

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}],{n}],Select[Tuples[#],UnsameQ@@#&]=={}&]]],{n,0,5}]

a(n) = A001434(n) - A137917(n).

a(25) onwards from Andrew Howroyd, Feb 02 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

Gus Wiseman, Feb 18 2024

nonn

			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}

Andrew Howroyd, Table of n, a(n) for n = 0..50

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.

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}]

\\ 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

a(8) onwards from Andrew Howroyd, Feb 19 2024

A054547 Triangular array giving number of labeled digraphs on n unisolated nodes and k=0..n*(n-1) arcs.

Original entry on oeis.org

0, 0, 2, 1, 0, 0, 12, 20, 15, 6, 1, 0, 0, 12, 140, 435, 768, 920, 792, 495, 220, 66, 12, 1, 0, 0, 0, 240, 2520, 11604, 34150, 73560, 123495, 166860, 184426, 167900, 125965, 77520, 38760, 15504, 4845, 1140, 190, 20, 1

Offset: 1

Vladeta Jovovic, Apr 09 2000

			Triangle T(n,k) begins:
  [0],
  [0,2,1],
  [0,0,12,20,15,6,1],
  [0,0,12,140,435,768,920,792,495,220,66,12,1],
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..2680 (rows 1..20)

Row sums are A054545.
Column sums are A121252.
The unlabeled version is A350908.
Cf. A054548 (graphs), A062735, A123554.

row(n) = {Vecrev(sum(i=0, n, (-1)^(n-i)*binomial(n,i)*(1 + 'y)^(i*(i-1))), n*(n-1)+1)}
{ for(n=1, 6, print(row(n))) } \\ Andrew Howroyd, Jan 28 2022

T(n, k) = Sum_{i=0..n} (-1)^(n-i)*binomial(n, i)*binomial(i*(i-1), k).

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

Gus Wiseman, Jan 23 2024

nonn

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

			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}

Andrew Howroyd, Table of n, a(n) for n = 0..50

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.

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}]

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

Gus Wiseman, Jan 23 2024

nonn

			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}

Andrew Howroyd, Table of n, a(n) for n = 0..500

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.

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}]

Euler transform of A005703 with A005703(1) = 0.
First differences of A134964.
a(n) = A002494(n) - A369202(n).

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

Gus Wiseman, Jan 19 2024

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

			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}

Andrew Howroyd, Table of n, a(n) for n = 0..1560 (rows 0..20)

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.

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]}]

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

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

cp's OEIS Frontend

A369145 Number of unlabeled loop-graphs with up to n vertices such that it is possible to choose a different vertex from each edge (choosable).

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Extensions

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

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Mathematica

Formula

A369193 Number of labeled simple graphs with n vertices and at most as many edges as covered (non-isolated) vertices.

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Mathematica

Formula

A001862 Number of forests of least height with n nodes.

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Mathematica

Formula

Extensions

A369201 Number of unlabeled simple graphs with n vertices and n edges such that it is not possible to choose a different vertex from each edge (non-choosable).

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Mathematica

Formula

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

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Mathematica

PARI

Extensions

A054547 Triangular array giving number of labeled digraphs on n unisolated nodes and k=0..n*(n-1) arcs.

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PARI

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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).

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