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 1-10 of 13 results. Next

A006649 Number of graphs with n nodes, n edges and no isolated vertices.

Original entry on oeis.org

1, 0, 0, 1, 2, 5, 15, 41, 124, 369, 1132, 3491, 10984, 34768, 111514, 360560, 1176797, 3870389, 12829765, 42829894, 143980892, 487227611, 1659499566, 5688046485, 19617965938, 68078878268, 237694501644, 834946053269, 2950683815028, 10490767818951, 37524169403930
Offset: 0

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).

Links

Crossrefs

Cf. A001434, A008406, A048179, A368599.

Programs

  • PARI
    a(n) = polcoef(G(n, O(x*x^n)) - if(n, G(n-1, O(x*x^n))), n) \\ G defined in A008406. - Andrew Howroyd, Jan 09 2024

Formula

a(n) = A008406(n,n) - A008406(n-1, n) for n > 0. - Andrew Howroyd, Jan 09 2024

Extensions

More terms from Vladeta Jovovic, Mar 02 2008
a(0)-a(2) prepended and a(27) and beyond from Andrew Howroyd, Jan 09 2024

A014068 a(n) = binomial(n*(n+1)/2, n).

Original entry on oeis.org

1, 1, 3, 20, 210, 3003, 54264, 1184040, 30260340, 886163135, 29248649430, 1074082795968, 43430966148115, 1917283000904460, 91748617512913200, 4730523156632595024, 261429178502421685800, 15415916972482007401455, 966121413245991846673830, 64123483527473864490450300
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Product of next n numbers divided by product of first n numbers. E.g., a(4) = (7*8*9*10)/(1*2*3*4)= 210. - Amarnath Murthy, Mar 22 2004
Also the number of labeled loop-graphs with n vertices and n edges. The covering case is A368597. - Gus Wiseman, Jan 25 2024

Examples

			From _Gus Wiseman_, Jan 25 2024: (Start)
The a(0) = 1 through a(3) = 20 loop-graph edge-sets (loops shown as singletons):
  {}  {{1}}  {{1},{2}}    {{1},{2},{3}}
             {{1},{1,2}}  {{1},{2},{1,2}}
             {{2},{1,2}}  {{1},{2},{1,3}}
                          {{1},{2},{2,3}}
                          {{1},{3},{1,2}}
                          {{1},{3},{1,3}}
                          {{1},{3},{2,3}}
                          {{2},{3},{1,2}}
                          {{2},{3},{1,3}}
                          {{2},{3},{2,3}}
                          {{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}}
(End)

Links

Crossrefs

Cf. A000217, A022915, A135860, A135861, A135862.
Diagonal of A084546.
Without loops we have A116508, covering A367863, unlabeled A006649.
Allowing edges of any positive size gives A136556, covering A054780.
The covering case is A368597.
The unlabeled version is A368598, covering A368599.
The connected case is A368951.
A000666 counts unlabeled loop-graphs, covering A322700.
A006125 (shifted left) counts loop-graphs, covering A322661.
A006129 counts covering simple graphs, connected A001187.
A058891 counts set-systems, unlabeled A000612.
Cf. A000085, A000272, A057500, A333331, A368596, A368730.

Programs

  • Magma
    [Binomial(Binomial(n+1,2), n): n in [0..40]]; // G. C. Greubel, Feb 19 2022
  • Mathematica
    Binomial[First[#],Last[#]]&/@With[{nn=20},Thread[{Accumulate[ Range[ 0,nn]], Range[ 0,nn]}]] (* Harvey P. Dale, May 27 2014 *)
  • Python
    from math import comb
def A014068(n): return comb(comb(n+1,2),n) # Chai Wah Wu, Jul 14 2024
  • Sage
    [(binomial(binomial(n+1, n-1), n)) for n in range(20)] # Zerinvary Lajos, Nov 30 2009

Formula

For n >= 1, Product_{k=1..n} a(k) = A022915(n). - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 08 2001
For n > 0, a(n) = A022915(n)/A022915(n-1). - Gerald McGarvey, Jul 26 2004
a(n) = binomial(T(n+1), T(n)) where T(n) = the n-th triangular number. - Amarnath Murthy, Jul 14 2005
a(n) = binomial(binomial(n+2, n), n+1) for n >= -1. - Zerinvary Lajos, Nov 30 2009
From Peter Bala, Feb 27 2020: (Start)
a(p) == (p + 1)/2 ( mod p^3 ) for prime p >= 5 (apply Mestrovic, equation 37).
Conjectural: a(2*p) == p*(2*p + 1) ( mod p^4 ) for prime p >= 5. (End)
a(n) = A084546(n,n). - Gus Wiseman, Jan 25 2024
a(n) = [x^n] (1+x)^(n*(n+1)/2). - Vaclav Kotesovec, Aug 06 2025

A368597 Number of n-element sets of singletons or pairs of distinct elements of {1..n} with union {1..n}, or loop-graphs covering n vertices with n edges.

Original entry on oeis.org

1, 1, 3, 17, 150, 1803, 27364, 501015, 10736010, 263461265, 7283725704, 223967628066, 7581128184175, 280103206674480, 11216492736563655, 483875783716549277, 22371631078155742764, 1103548801569848115255, 57849356643299101021960, 3211439288584038922502820
Offset: 0

Views

Author

Gus Wiseman, Jan 04 2024

Keywords

Comments

It doesn't matter for this sequence whether we use loops such as {x,x} or half-loops such as {x}.

Examples

			The a(0) = 1 through a(3) = 17 set-systems:
  {}  {{1}}  {{1},{2}}    {{1},{2},{3}}
             {{1},{1,2}}  {{1},{2},{1,3}}
             {{2},{1,2}}  {{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}}

Links

Crossrefs

This is the covering case of A014068.
Allowing edges of any positive size gives A054780, covering case of A136556.
Allowing any number of edges gives A322661, connected A062740.
The case of just pairs is A367863, covering case of A116508.
The unlabeled version is A368599.
The version contradicting strict AOC is A368730.
The connected case is A368951.
A000085 counts set partitions into singletons or pairs.
A006129 counts covering graphs, connected A001187.
A058891 counts set-systems, unlabeled A000612.
A100861 counts set partitions into singletons or pairs by number of pairs.
A111924 counts set partitions into singletons or pairs by length.
Cf. A000272, A000666, A057500, A066383, A333331, A367869, A368596, A368600, A368928, A368951, A369199.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Subsets[Range[n],{1,2}], {n}],Union@@#==Range[n]&]],{n,0,5}]
  • PARI
    a(n) = sum(k=0, n, (-1)^(n-k) * binomial(n,k) * binomial(binomial(k+1,2), n)) \\ Andrew Howroyd, Jan 06 2024

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * binomial(binomial(k+1,2), n). - Andrew Howroyd, Jan 06 2024

Extensions

Terms a(7) and beyond from Andrew Howroyd, Jan 06 2024

A369194 Number of labeled loop-graphs covering n vertices with at most n edges.

Original entry on oeis.org

1, 1, 4, 23, 199, 2313, 34015, 606407, 12712643, 306407645, 8346154699, 253476928293, 8490863621050, 310937199521774, 12356288017546937, 529516578044589407, 24339848939829286381, 1194495870124420574751, 62332449791125883072149, 3446265450868329833016605
Offset: 0

Views

Author

Gus Wiseman, Jan 17 2024

Keywords

Comments

Row-sums of left portion of A369199.

Examples

			The a(0) = 1 through a(3) = 23 loop-graphs (loops shown as singletons):
  {}  {{1}}  {{1,2}}      {{1},{2,3}}
             {{1},{2}}    {{2},{1,3}}
             {{1},{1,2}}  {{3},{1,2}}
             {{2},{1,2}}  {{1,2},{1,3}}
                          {{1,2},{2,3}}
                          {{1},{2},{3}}
                          {{1,3},{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 minimal case is A001862, without loops A053530.
This is the covering case of A066383 and A369196, cf. A369192 and A369193.
The case of equality is A368597, without loops A367863.
The version without loops is A369191.
The connected case is A369197, without loops A129271.
The unlabeled version is A370169, equality A368599, non-covering A368598.
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.
A133686 counts choosable graphs, covering A367869.
A322661 counts covering loop-graphs, unlabeled A322700.
A367867 counts non-choosable graphs, covering A367868.
A368927 counts choosable loop-graphs, covering A369140.
A369141 counts non-choosable loop-graphs, covering A369142.
Cf. A000169, A000272, A000666, A003465, A005703, A006649, A057500, A062740, A116508, A367862, A367916.

Programs

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

Formula

Inverse binomial transform of A369196.

A368984 Number of graphs with loops (symmetric relations) on n unlabeled vertices in which each connected component has an equal number of vertices and edges.

Original entry on oeis.org

1, 1, 2, 5, 12, 29, 75, 191, 504, 1339, 3610, 9800, 26881, 74118, 205706, 573514, 1606107, 4513830, 12727944, 35989960, 102026638, 289877828, 825273050, 2353794251, 6724468631, 19239746730, 55123700591, 158133959239, 454168562921, 1305796834570, 3758088009136
Offset: 0

Views

Author

Andrew Howroyd, Jan 11 2024

Keywords

Comments

The graphs considered here can have loops but not parallel edges.
Also the number of unlabeled loop-graphs with n edges and n vertices such that it is possible to choose a different vertex from each edge. - Gus Wiseman, Jan 25 2024

Examples

			Representatives of the a(3) = 5 graphs are:
   {{1,2}, {1,3}, {2,3}},
   {{1}, {1,2}, {1,3}},
   {{1}, {1,2}, {2,3}},
   {{1}, {2}, {2,3}},
   {{1}, {2}, {3}}.
The graph with 4 vertices and edges {{1}, {2}, {1,2}, {3,4}} is included by A368599 but not by this sequence.

Links

Crossrefs

The case of a unique choice is A000081.
Without loops we have A137917, labeled A137916.
The labeled version appears to be A333331.
Without the choice condition we have A368598, covering A368599.
The complement is counted by A368835, labeled A368596 (covering A368730).
Row sums of A368926, labeled A368924.
The connected case is A368983.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A000666 counts unlabeled loop-graphs, covering A322700.
A006125 counts simple graphs, unlabeled A000088.
A006129 counts covering graphs, connected A001187, unlabeled A002494.
A322661 counts labeled covering loop-graphs, connected A062740.
Cf. A014068, A057500, A116508, A129271, A133686, A367863, A367869, A367902, A368597, A368601, A368836.

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}],{n}],Length[Select[Tuples[#],UnsameQ@@#&]]!=0&]]],{n,0,5}] (* Gus Wiseman, Jan 25 2024 *)

Formula

Euler transform of A368983.

A368598 Number of non-isomorphic n-element sets of singletons or pairs of elements of {1..n}, or unlabeled loop-graphs with n edges and up to n vertices.

Original entry on oeis.org

1, 1, 2, 6, 17, 52, 173, 585, 2064, 7520, 28265, 109501, 437394, 1799843, 7629463, 33302834, 149633151, 691702799, 3287804961, 16058229900, 80533510224, 414384339438, 2185878202630, 11811050484851, 65318772618624, 369428031895444, 2135166786135671, 12601624505404858
Offset: 0

Views

Author

Gus Wiseman, Jan 05 2024

Keywords

Comments

It doesn't matter for this sequence whether we use loops such as {x,x} or half-loops such as {x}.

Examples

			Non-isomorphic representatives of the a(0) = 1 through a(4) = 17 set-systems:
  {}  {{1}}  {{1},{2}}    {{1},{2},{3}}        {{1},{2},{3},{4}}
             {{1},{1,2}}  {{1},{2},{1,2}}      {{1},{2},{3},{1,2}}
                          {{1},{2},{1,3}}      {{1},{2},{3},{1,4}}
                          {{1},{1,2},{1,3}}    {{1},{2},{1,2},{1,3}}
                          {{1},{1,2},{2,3}}    {{1},{2},{1,2},{3,4}}
                          {{1,2},{1,3},{2,3}}  {{1},{2},{1,3},{1,4}}
                                               {{1},{2},{1,3},{2,3}}
                                               {{1},{2},{1,3},{2,4}}
                                               {{1},{3},{1,2},{2,4}}
                                               {{1},{1,2},{1,3},{1,4}}
                                               {{1},{1,2},{1,3},{2,3}}
                                               {{1},{1,2},{1,3},{2,4}}
                                               {{1},{1,2},{2,3},{3,4}}
                                               {{2},{1,2},{1,3},{1,4}}
                                               {{4},{1,2},{1,3},{2,3}}
                                               {{1,2},{1,3},{1,4},{2,3}}
                                               {{1,2},{1,3},{2,4},{3,4}}

Links

Crossrefs

For any number of edges of any size we have A000612, covering A055621.
For any number of edges we have A000666, A054921, A322700.
The labeled version is A014068.
Counting by weight gives A320663, or A339888 with loops {x,x}.
The covering case is A368599.
For edges of any size we have A368731, covering A368186.
Row sums of A368836.
A000085 counts set partitions into singletons or pairs.
A001515 counts length-n set partitions into singletons or pairs.
A100861 counts set partitions into singletons or pairs by number of pairs.
A111924 counts set partitions into singletons or pairs by length.
Cf. A001434, A007716, A007717, A058891, A070166, A122848, A124059, A283877, A302545, A322661, A339741, A339887, A370168.

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],{1,2}],{n}]]],{n,0,5}]
  • PARI
    a(n) = polcoef(G(n, O(x*x^n)), n) \\ G defined in A070166. - Andrew Howroyd, Jan 09 2024

Formula

a(n) = A070166(n, n). - Andrew Howroyd, Jan 09 2024

Extensions

Terms a(7) and beyond from Andrew Howroyd, Jan 09 2024

A070166 Irregular triangle read by rows giving T(n,k) = number of rooted graphs on n + 1 nodes with k edges (n >= 0, 0 <= k <= n(n-1)/2).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 1, 2, 4, 6, 4, 2, 1, 1, 2, 5, 11, 17, 18, 17, 11, 5, 2, 1, 1, 2, 5, 13, 29, 52, 76, 94, 94, 76, 52, 29, 13, 5, 2, 1, 1, 2, 5, 14, 35, 83, 173, 308, 487, 666, 774, 774, 666, 487, 308, 173, 83, 35, 14, 5, 2, 1, 1, 2, 5, 14, 37, 98, 252, 585, 1239, 2396, 4135, 6340
Offset: 0

Views

Author

Vladeta Jovovic and Eric W. Weisstein, Apr 23 2002

Keywords

Comments

T(n,k) is also the number of graphs with n nodes and k edges which may contain loops. (Delete the root node and change every edge leading to it into a loop.)
T(n,k) is also the number of symmetric relations with n points and k relations.

Examples

			Triangle begins:
1;
1, 1;
1, 2, 2, 1;
1, 2, 4, 6, 4, 2, 1;
1, 2, 5, 11, 17, 18, 17, 11, 5, 2, 1; <- gives either the numbers of rooted graphs on 5 nodes, or the numbers of graphs with loops on 4 nodes; with from 0 to 10 edges
1, 2, 5, 13, 29, 52, 76, 94, 94, 76, 52, 29, 13, 5, 2, 1;
...

References

  • E. Palmer and F. Harary, Graphical Enumeration, Academic Press, 1973.

Links

Crossrefs

Row sums are A000666.
Cf. A054921, A008406, A283755, A322114, A368598, A368599.

Programs

  • Mathematica
    Join[{{1},{1,1}},CoefficientList[Table[CycleIndex[Join[PairGroup[SymmetricGroup[n]],Permutations[Range[Binomial[n,2]+1,Binomial[n,2]+n]],2],s]/.Table[s[i]->1+x^i,{i,1,n^2-n}],{n,2,7}],x]]//Grid  (* Geoffrey Critzer, Oct 01 2012 *)
permcount[v_] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t*k; s += t]; s!/m];
edges[v_, t_] := Product[Product[g = GCD[v[[i]], v[[j]]]; t[v[[i]]*v[[j]]/g]^g, {j, 1, i - 1}], {i, 2, Length[v]}]*Product[c = v[[i]]; t[c]^Quotient[c + 1, 2]*If[OddQ[c], 1, t[c/2]], {i, 1, Length[v]}];
row[n_] := Module[{s = 0}, Do[s += permcount[p]*edges[p, 1 + x^# &], {p, IntegerPartitions[n]}]; s/n!] // Expand // CoefficientList[#, x] &
row /@ Range[0, 7] // Flatten (* Jean-François Alcover, Jan 07 2021, after Andrew Howroyd *)
  • PARI
    permcount(v) = {my(m=1,s=0,k=0,t); for(i=1,#v,t=v[i]; k=if(i>1&&t==v[i-1],k+1,1); m*=t*k;s+=t); s!/m}
edges(v, t) = {prod(i=2, #v, prod(j=1, i-1, my(g=gcd(v[i], v[j])); t(v[i]*v[j]/g)^g )) * prod(i=1, #v, my(c=v[i]); t(c)^((c+1)\2)*if(c%2, 1, t(c/2)))}
G(n, A=0) = {my(s=0); forpart(p=n, s+=permcount(p)*edges(p, i->1+x^i+A)); s/n!}
{ for(n=0, 7, print(Vecrev(G(n)))) } \\ Andrew Howroyd, Oct 23 2019, updated Jan 09 2024

Extensions

Offset changed by Andrew Howroyd, Oct 23 2019

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

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

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

Views

Author

Gus Wiseman, Feb 18 2024

Keywords

Examples

			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}

Links

Crossrefs

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.

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}],{0,n}], Union@@#==Range[n]&]]],{n,0,5}]
  • PARI
    \\ 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

Extensions

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