A367868 -id:A367868 - cp's OEIS frontend

A137916 Number of n-node labeled graphs whose components are unicyclic graphs.

1, 0, 0, 1, 15, 222, 3670, 68820, 1456875, 34506640, 906073524, 26154657270, 823808845585, 28129686128940, 1035350305641990, 40871383866109888, 1722832666898627865, 77242791668604946560, 3670690919234354407000, 184312149879830557190940, 9751080154504005703189791

Offset: 0

Washington Bomfim, Feb 22 2008

Also the number of labeled simple graphs with n vertices and n edges such that it is possible to choose a different vertex from each edge. The version without the choice condition is A116508, covering A367863. - Gus Wiseman, Jan 25 2024

			a(6) = 3670 because A057500(6) = 3660 and two triangles can be labeled in 10 ways.
From _Gus Wiseman_, Jan 25 2024: (Start)
The a(0) = 1 through a(4) = 15 simple graphs:
  {}  .  .  {12,13,23}  {12,13,14,23}
                        {12,13,14,24}
                        {12,13,14,34}
                        {12,13,23,24}
                        {12,13,23,34}
                        {12,13,24,34}
                        {12,14,23,24}
                        {12,14,23,34}
                        {12,14,24,34}
                        {12,23,24,34}
                        {13,14,23,24}
                        {13,14,23,34}
                        {13,14,24,34}
                        {13,23,24,34}
                        {14,23,24,34}
(End)

V. F. Kolchin, Random Graphs. Encyclopedia of Mathematics and Its Applications 53. Cambridge Univ. Press, Cambridge, 1999.

Alois P. Heinz, Table of n, a(n) for n = 0..386 (terms n = 151..200 from Andrew Howroyd)
Eric Weisstein's World of Mathematics, Pseudoforest.
Wikipedia, Pseudoforest.

The connected case is A057500.
Row sums of A106239.
The unlabeled version is A137917.
Diagonal of A144228.
The version with loops appears to be A333331, unlabeled A368984.
Column k = 0 of A368924.
The complement is counted by A369143, unlabeled A369201, covering A369144.
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 simple graphs, covering A367869.
A140637 counts unlabeled non-choosable graphs, covering A369202.
A367867 counts non-choosable graphs, covering A367868.
Cf. A001434, A006649, A053530, A116508, A129271, A134964, A140638, A367862, A367863, A369191.

cy:= proc(n) option remember;
       binomial(n-1, 2)*add((n-3)!/(n-2-t)!*n^(n-2-t), t=1..n-2)
     end:
T:= proc(n,k) option remember; `if`(k=0, 1, `if`(k<0 or n T(n,n):
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 15 2008

nn = 20; t = Sum[n^(n - 1) x^n/n!, {n, 1, nn}]; Drop[Range[0, nn]! CoefficientList[Series[Exp[Log[1/(1 - t)]/2 - t/2 - t^2/4], {x, 0, nn}], x], 1] (* Geoffrey Critzer, Jan 24 2012 *)
Table[Length[Select[Subsets[Subsets[Range[n],{2}],{n}],Length[Select[Tuples[#],UnsameQ@@#&]]!=0&]],{n,0,5}] (* Gus Wiseman, Jan 25 2024 *)

A057500(p) = (p-1)! * p^p /2 * sum(k = 3,p, 1/(p^k*(p-k)!)); /* Vladeta Jovovic, A057500. */
F(n,N) = { my(s = 0, K, D, Mc); forpart(P = n, D = Set(P); K = vector(#D);
for(i=1, #D, K[i] = #select(x->x == D[i], Vec(P)));
Mc = n!/prod(i=1,#D, K[i]!);
s += Mc * prod(i = 1, #D, A057500(D[i])^K[i] / ( D[i]!^K[i])) , [3, n], [N, N]); s };
a(n)= {my(N); sum(N = 1, n, F(n,N))};

seq(n)={my(w=lambertw(-x+O(x*x^n))); Vec(serlaplace(exp(-log(1+w)/2 + w/2 - w^2/4)))} \\ Andrew Howroyd, May 18 2021

a(n) = Sum_{N = 1..n} ((n!/N!) * Sum_{n_1 + n_2 + ... + n_N = n} Product_{i = 1..N} ( A057500(n_i) / n_i! ) ). [V. F. Kolchin p. 31, (1.4.2)] replacing numerator terms n_i^(n_i-2) by A057500(n_i).
a(n) = A144228(n,n). - Alois P. Heinz, Sep 15 2008
E.g.f.: exp(B(T(x))) where B(x) = (log(1/(1-x))-x-x^2/2)/2 and T(x) is the e.g.f. for A000169 (labeled rooted trees). - Geoffrey Critzer, Jan 24 2012
a(n) ~ 2^(-1/4)*exp(-3/4)*GAMMA(3/4)*n^(n-1/4)/sqrt(Pi) * (1-7*Pi/(12*GAMMA(3/4)^2*sqrt(n))). - Vaclav Kotesovec, Aug 16 2013
E.g.f.: exp(B(x)) where B(x) is the e.g.f. of A057500. - Andrew Howroyd, May 18 2021

a(0)=1 prepended by Andrew Howroyd, May 18 2021

A367901 Number of sets of subsets of {1..n} contradicting a strict version of the axiom of choice.

Original entry on oeis.org

1, 2, 9, 195, 63765, 4294780073, 18446744073639513336, 340282366920938463463374607341656713953, 115792089237316195423570985008687907853269984665640564039457583610129753447747

Offset: 0

Gus Wiseman, Dec 05 2023

nonn

The axiom of choice says that, given any set of nonempty sets Y, it is possible to choose a set containing an element from each. The strict version requires this set to have the same cardinality as Y, meaning no element is chosen more than once.

			The a(2) = 9 sets of sets:
  {{}}
  {{},{1}}
  {{},{2}}
  {{},{1,2}}
  {{},{1},{2}}
  {{},{1},{1,2}}
  {{},{2},{1,2}}
  {{1},{2},{1,2}}
  {{},{1},{2},{1,2}}

Wikipedia, Axiom of choice.

The version for simple graphs is A367867, covering A367868.
The complement is counted by A367902, no singletons A367770, ranks A367906.
The version without empty edges is A367903, ranks A367907.
For a unique choice (instead of none) we have A367904, ranks A367908.
A000372 counts antichains, covering A006126, nonempty A014466.
A003465 counts covering set-systems, unlabeled A055621.
A058891 counts set-systems, unlabeled A000612.
A059201 counts covering T_0 set-systems.
A323818 counts covering connected set-systems, unlabeled A323819.
A326031 gives weight of the set-system with BII-number n.
Cf. A007716, A083323, A092918, A102896, A283877, A306445, A355739, A355740, A367769, A367862, A367905.

Table[Length[Select[Subsets[Subsets[Range[n]]], Select[Tuples[#],UnsameQ@@#&]=={}&]],{n,0,3}]

a(n) = 2^2^n - A367902(n). - Christian Sievers, Aug 01 2024

a(5)-a(8) from Christian Sievers, Aug 01 2024

A367862 Number of n-vertex labeled simple graphs with the same number of edges as covered vertices.

Original entry on oeis.org

1, 1, 1, 2, 20, 308, 5338, 105298, 2366704, 60065072, 1702900574, 53400243419, 1836274300504, 68730359299960, 2782263907231153, 121137565273808792, 5645321914669112342, 280401845830658755142, 14788386825536445299398, 825378055206721558026931, 48604149005046792753887416

Offset: 0

Gus Wiseman, Dec 07 2023

nonn

Unlike the connected case (A057500), these graphs may have more than one cycle; for example, the graph {{1,2},{1,3},{1,4},{2,3},{2,4},{5,6}} has multiple cycles.

			Non-isomorphic representatives of the a(4) = 20 graphs:
  {}
  {{1,2},{1,3},{2,3}}
  {{1,2},{1,3},{1,4},{2,3}}
  {{1,2},{1,3},{2,4},{3,4}}

Andrew Howroyd, Table of n, a(n) for n = 0..200

The connected case is A057500, unlabeled A001429.
Counting all vertices (not just covered) gives A116508.
The covering case is A367863, unlabeled A006649.
For set-systems we have A367916, ranks A367917.
A001187 counts connected graphs, A001349 unlabeled.
A003465 counts covering set-systems, unlabeled A055621, ranks A326754.
A006125 counts graphs, A000088 unlabeled.
A006129 counts covering graphs, A002494 unlabeled.
A058891 counts set-systems, unlabeled A000612, without singletons A016031.
A059201 counts covering T_0 set-systems, unlabeled A319637, ranks A326947.
A133686 = graphs satisfy strict AoC, connected A129271, covering A367869.
A143543 counts simple labeled graphs by number of connected components.
A323818 counts connected set-systems, unlabeled A323819, ranks A326749.
A367867 = graphs contradict strict AoC, connected A140638, covering A367868.
Cf. A006126, A316983, A321405, A326754, A326870, A326877, A367770, A367901, A367902, A367903.

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

\\ Here b(n) is A367863(n)
b(n) = sum(k=0, n, (-1)^(n-k) * binomial(n,k) * binomial(binomial(k,2), n))
a(n) = sum(k=0, n, binomial(n,k) * b(k)) \\ Andrew Howroyd, Dec 29 2023

Binomial transform of A367863.

Terms a(8) and beyond from Andrew Howroyd, Dec 29 2023

A140638 Number of connected graphs on n labeled nodes that contain at least two cycles.

Original entry on oeis.org

0, 0, 0, 7, 381, 21748, 1781154, 249849880, 66257728763, 34495508486976, 35641629989151608, 73354595357480683904, 301272202621204113362497, 2471648811029413368450098688, 40527680937730440155535277704046, 1328578958335783199341353852258282496

Offset: 1

Washington Bomfim, May 21 2008

nonn

These are the connected graphs that are neither trees nor unicyclic.

Also connected non-choosable graphs covering n vertices, where a graph is choosable iff it is possible to choose a different vertex from each edge. The unlabeled version is A140636. The complement is counted by A129271. - Gus Wiseman, Feb 20 2024

J. Riordan, An Introduction to Combinatorial Analysis, Dover, 2002, p. 2.

Andrew Howroyd, Table of n, a(n) for n = 1..50

The unlabeled version is A140636.
Cf. A000272 (trees), A001187 (connected graphs), A057500 (connected unicyclic graphs).
The complement is counted by A129271, unlabeled A005703.
The non-connected complement is A133686, covering A367869.
The non-connected version is A367867, unlabeled A140637.
The non-connected covering version is A367868.
A006125 counts graphs, A000088 unlabeled.
A006129 counts covering graphs, A002494 unlabeled.
A143543 counts simple labeled graphs by number of connected components.
Cf. A001349, A116508, A355740, A367769, A367863, A367901, A367903, A370317.

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}]],Union@@#==Range[n]&&Length[csm[#]]<=1&&Select[Tuples[#],UnsameQ@@#&]=={}&]],{n,0,5}] (* Gus Wiseman, Feb 19 2024 *)

seq(n)={my(A=O(x*x^n), t=-lambertw(-x + A)); Vec(serlaplace( log(sum(k=0, n, 2^binomial(k, 2)*x^k/k!, A)) - log(1/(1-t))/2 - t/2 + 3*t^2/4), -n)} \\ Andrew Howroyd, Jan 15 2022

a(n) = A001187(n) - A129271(n).

a(n) = A001187(n) - A000272(n) - A057500(n).

Definition clarified by Andrew Howroyd, Jan 15 2022

A370636 Number of subsets of {1..n} such that it is possible to choose a different binary index of each element.

Original entry on oeis.org

1, 2, 4, 7, 14, 24, 39, 61, 122, 203, 315, 469, 676, 952, 1307, 1771, 3542, 5708, 8432, 11877, 16123, 21415, 27835, 35757, 45343, 57010, 70778, 87384, 106479, 129304, 155802, 187223, 374446, 588130, 835800, 1124981, 1456282, 1841361, 2281772, 2791896, 3367162

Offset: 0

Gus Wiseman, Mar 08 2024

nonn

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

			The a(0) = 1 through a(4) = 14 subsets:
  {}  {}   {}     {}     {}
      {1}  {1}    {1}    {1}
           {2}    {2}    {2}
           {1,2}  {3}    {3}
                  {1,2}  {4}
                  {1,3}  {1,2}
                  {2,3}  {1,3}
                         {1,4}
                         {2,3}
                         {2,4}
                         {3,4}
                         {1,2,4}
                         {1,3,4}
                         {2,3,4}

Wikipedia, Axiom of choice.
Gus Wiseman, Statistics, classes, and transformations of standard compositions

Simple graphs of this type are counted by A133686, covering A367869.
Unlabeled graphs of this type are counted by A134964, complement A140637.
Simple graphs not of this type are counted by A367867, covering A367868.
Set systems of this type are counted by A367902, ranks A367906.
Set systems not of this type are counted by A367903, ranks A367907.
Set systems uniquely of this type are counted by A367904, ranks A367908.
Unlabeled multiset partitions of this type are A368098, complement A368097.
A version for MM-numbers of multisets is A368100, complement A355529.
Factorizations are counted by A368414/A370814, complement A368413/A370813.
For prime indices we have A370582, differences A370586.
The complement for prime indices is A370583, differences A370587.
The complement is A370637, differences A370589, without ones A370643.
The case of a unique choice is A370638, maxima A370640, differences A370641.
First differences are A370639.
The minimal case of the complement is A370642, without ones A370644.
A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.
A058891 counts set-systems, A003465 covering, A323818 connected.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
A326031 gives weight of the set-system with BII-number n.
Cf. A000612, A326702, A355739, A355740, A367772, A367905, A367909, A367912, A368095, A368109.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Table[Length[Select[Subsets[Range[n]], Select[Tuples[bpe/@#],UnsameQ@@#&]!={}&]],{n,0,10}]

a(2^n - 1) = A367902(n).

Partial sums of A370639.

a(19)-a(40) from Alois P. Heinz, Mar 09 2024

A369141 Number of labeled loop-graphs covering a subset of {1..n} such that it is not possible to choose a different vertex from each edge (non-choosable).

Original entry on oeis.org

0, 0, 1, 25, 710, 29394, 2051522, 267690539, 68705230758, 35184059906570, 36028789310419722, 73786976083150073999, 302231454897259573627852, 2475880078570549574773324062, 40564819207303333310731978895956, 1329227995784915872613854321228773937

Offset: 0

Gus Wiseman, Jan 20 2024

nonn

Also labeled loop-graphs having at least one connected component containing more edges than vertices.

			The a(0) = 0 through a(3) = 25 loop-graphs (loops shown as singletons):
  .  .  {{1},{2},{1,2}}  {{1},{2},{1,2}}
                         {{1},{3},{1,3}}
                         {{2},{3},{2,3}}
                         {{1},{2},{3},{1,2}}
                         {{1},{2},{3},{1,3}}
                         {{1},{2},{3},{2,3}}
                         {{1},{2},{1,2},{1,3}}
                         {{1},{2},{1,2},{2,3}}
                         {{1},{2},{1,3},{2,3}}
                         {{1},{3},{1,2},{1,3}}
                         {{1},{3},{1,2},{2,3}}
                         {{1},{3},{1,3},{2,3}}
                         {{2},{3},{1,2},{1,3}}
                         {{2},{3},{1,2},{2,3}}
                         {{2},{3},{1,3},{2,3}}
                         {{1},{1,2},{1,3},{2,3}}
                         {{2},{1,2},{1,3},{2,3}}
                         {{3},{1,2},{1,3},{2,3}}
                         {{1},{2},{3},{1,2},{1,3}}
                         {{1},{2},{3},{1,2},{2,3}}
                         {{1},{2},{3},{1,3},{2,3}}
                         {{1},{2},{1,2},{1,3},{2,3}}
                         {{1},{3},{1,2},{1,3},{2,3}}
                         {{2},{3},{1,2},{1,3},{2,3}}
                         {{1},{2},{3},{1,2},{1,3},{2,3}}

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

Without the choice condition we have A006125, unlabeled A000088.
The case of a unique choice is A088957, unlabeled A087803.
The case without loops is A367867, covering A367868.
For edges of any positive size we have A367903, complement A367902.
For exactly n edges we have A368596, complement A333331 (maybe).
The complement is counted by A368927, covering A369140.
The covering case is A369142.
For n edges and no loops we have A369143, covering A369144.
The unlabeled version is A369146 (covering A369147), complement A369145.
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.
A133686 counts choosable graphs, covering A367869.
A322661 counts labeled covering loop-graphs, unlabeled A322700.
Cf. A000272, A000666, A006649, A062740, A116508, A137916, A368924, A368984, A369194.

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

Binomial transform of A369142.

a(n) = A006125(n + 1) - A368927(n). - Andrew Howroyd, Feb 02 2024

a(6) onwards from Andrew Howroyd, Feb 02 2024

A368596 Number of n-element sets of singletons or pairs of distinct elements of {1..n}, or loop graphs with n edges, such that it is not possible to choose a different element from each.

Original entry on oeis.org

0, 0, 0, 3, 66, 1380, 31460, 800625, 22758918, 718821852, 25057509036, 957657379437, 39878893266795, 1799220308202603, 87502582432459584, 4566246347310609247, 254625879822078742956, 15115640124974801925030, 952050565540607423524658, 63425827673509972464868323

Offset: 0

Gus Wiseman, Jan 04 2024

nonn

The axiom of choice says that, given any set of nonempty sets Y, it is possible to choose a set containing an element from each. The strict version requires this set to have the same cardinality as Y, meaning no element is chosen more than once.

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

Andrew Howroyd, Table of n, a(n) for n = 0..200

The version without the choice condition is A014068, covering A368597.
The complement appears to be A333331.
For covering pairs we have A367868.
Allowing edges of any positive size gives A368600, any length A367903.
The covering case is A368730.
The unlabeled version is A368835.
A000085 counts set partitions into singletons or pairs.
A006125 counts graphs, unlabeled A000088.
A058891 counts set-systems (without singletons A016031), unlabeled A000612.
A100861 counts set partitions into singletons or pairs by number of pairs.
A111924 counts set partitions into singletons or pairs by length.
A322661 counts covering half-loop-graphs, connected A062740.
A369141 counts non-choosable loop-graphs, covering A369142.
A369146 counts unlabeled non-choosable loop-graphs, covering A369147.
Cf. A000272, A000666, A057500, A129271, A133686, A367769, A367863, A367867, A367869, A367901, A367907, A368097, A369199.

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

Terms a(7) and beyond from Andrew Howroyd, Jan 10 2024

A367769 Number of finite sets of nonempty non-singleton subsets of {1..n} contradicting a strict version of the axiom of choice.

Original entry on oeis.org

0, 0, 0, 1, 1490, 67027582, 144115188036455750, 1329227995784915872903806998967001298, 226156424291633194186662080095093570025917938800079226639565284090686126876

Offset: 0

Gus Wiseman, Dec 05 2023

The axiom of choice says that, given any set of nonempty sets Y, it is possible to choose a set containing an element from each. The strict version requires this set to have the same cardinality as Y, meaning no element is chosen more than once.

Includes all set-systems with more edges than covered vertices, but this condition is not sufficient.

			The a(3) = 1 set-system is: {{1,2},{1,3},{2,3},{1,2,3}}.

Wikipedia, Axiom of choice.

Set-systems without singletons are counted by A016031, covering A323816.
The complement is A367770, with singletons allowed A367902 (ranks A367906).
The version for simple graphs is A367867, covering A367868.
The version allowing singletons and empty edges is A367901.
The version allowing singletons is A367903, ranks A367907.
A000372 counts antichains, covering A006126, nonempty A014466.
A003465 counts covering set-systems, unlabeled A055621.
A058891 counts set-systems, unlabeled A000612.
A059201 counts covering T_0 set-systems.
A323818 counts covering connected set-systems.
Cf. A007716, A092918, A102896, A283877, A306445, A326031, A355739, A355740, A355741, A367904, A367905.

Table[Length[Select[Subsets[Select[Subsets[Range[n]], Length[#]>1&]], Select[Tuples[#], UnsameQ@@#&]=={}&]], {n,0,3}]

a(n) = 2^(2^n-n-1) - A367770(n) = A016031(n+1) - A367770(n). - Christian Sievers, Jul 28 2024

a(6)-a(8) from Christian Sievers, Jul 28 2024

A370637 Number of subsets of {1..n} such that it is not possible to choose a different binary index of each element.

Original entry on oeis.org

0, 0, 0, 1, 2, 8, 25, 67, 134, 309, 709, 1579, 3420, 7240, 15077, 30997, 61994, 125364, 253712, 512411, 1032453, 2075737, 4166469, 8352851, 16731873, 33497422, 67038086, 134130344, 268328977, 536741608, 1073586022, 2147296425, 4294592850, 8589346462, 17179033384

Offset: 0

Gus Wiseman, Mar 08 2024

nonn

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

			The a(0) = 0 through a(5) = 8 subsets:
  .  .  .  {1,2,3}  {1,2,3}    {1,2,3}
                    {1,2,3,4}  {1,4,5}
                               {1,2,3,4}
                               {1,2,3,5}
                               {1,2,4,5}
                               {1,3,4,5}
                               {2,3,4,5}
                               {1,2,3,4,5}

Wikipedia, Axiom of choice.
Gus Wiseman, Statistics, classes, and transformations of standard compositions

Simple graphs not of this type are counted by A133686, covering A367869.
Unlabeled graphs of this type are counted by A140637, complement A134964.
Simple graphs of this type are counted by A367867, covering A367868.
Set systems not of this type are counted by A367902, ranks A367906.
Set systems of this type are counted by A367903, ranks A367907.
Set systems uniquely not of this type are counted by A367904, ranks A367908.
Unlabeled multiset partitions of this type are A368097, complement A368098.
A version for MM-numbers of multisets is A355529, complement A368100.
Factorizations are counted by A368413/A370813, complement A368414/A370814.
The complement for prime indices is A370582, differences A370586.
For prime indices we have A370583, differences A370587.
First differences are A370589.
The complement is counted by A370636, differences A370639.
The case without ones is A370643.
The version for a unique choice is A370638, maxima A370640, diffs A370641.
The minimal case is A370642, without ones A370644.
A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.
A058891 counts set-systems, A003465 covering, A323818 connected.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
A326031 gives weight of the set-system with BII-number n.
Cf. A000612, A072639, A355739, A355740, A367772, A367905, A367909, A367912, A368094, A368095, A368109.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Table[Length[Select[Subsets[Range[n]], Select[Tuples[bpe/@#],UnsameQ@@#&]=={}&]],{n,0,10}]

a(2^n - 1) = A367903(n).

Partial sums of A370589.

a(21)-a(34) from Alois P. Heinz, Mar 09 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

Gus Wiseman, Jan 17 2024

nonn

Row-sums of left portion of A369199.

			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}}

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.

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

Inverse binomial transform of A369196.

cp's OEIS Frontend

A137916 Number of n-node labeled graphs whose components are unicyclic graphs.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

Extensions

A367901 Number of sets of subsets of {1..n} contradicting a strict version of the axiom of choice.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A367862 Number of n-vertex labeled simple graphs with the same number of edges as covered vertices.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A140638 Number of connected graphs on n labeled nodes that contain at least two cycles.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A370636 Number of subsets of {1..n} such that it is possible to choose a different binary index of each element.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A369141 Number of labeled loop-graphs covering a subset of {1..n} such that it is not possible to choose a different vertex from each edge (non-choosable).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A368596 Number of n-element sets of singletons or pairs of distinct elements of {1..n}, or loop graphs with n edges, such that it is not possible to choose a different element from each.