A367867 -id:A367867 - cp's OEIS frontend

A140637 Number of unlabeled graphs of positive excess with n nodes.

0, 0, 0, 2, 15, 110, 936, 12073, 273972, 12003332, 1018992968, 165091159269, 50502031331411, 29054155657134165, 31426485969804026075, 64001015704527557101231, 245935864153532932681481794, 1787577725145611700547871854870, 24637809253125004524383007473440146

Offset: 1

Washington Bomfim, May 21 2008

nonn

We can find in "The Birth of the Giant Component" p. 53, see the link, the following: "The excess of a graph or multigraph is the number of edges plus the number of acyclic components, minus the number of vertices."
If G has just one complex component with 4 nodes, the "non-complex part" of G can be,
a) One forest of order 4. There are 6 forests, so 2*6=12 graphs.
b) One triangle and one isolated vertex, or 2*1=2 graphs.
c) One unicyclic graph of order 4, so 2*2=4 graphs.
Also the number of unchoosable unlabeled graphs with up to n vertices, where a graph is choosable iff it is possible to choose a different vertex from each edge. The labeled version is A367867, covering A367868, connected A140638. - Gus Wiseman, Feb 13 2024

			Below we show that a(8) = 12073. Note that A140636(n) is the number of connected graphs of positive excess with n nodes.
Let G be a disconnected graph of positive excess with 8 nodes. In this case, G has one or two complex components. We have 3 graphs of order 8 with two complex components. One of those graphs is depicted in the figure below:
  O---O...O---O
  |\..|...|\./|
  |.\.|...|.X.|
  |..\|...|/.\|
  O---O...O---O
If G has one complex component with 5 nodes, the non-complex part of G can be,
a) One forest of order 3. There are 3 forests, so A140636(5) * 3 = 39 graphs.
b) One triangle, so A140636(5) = 13 graphs.
If G has one complex component with 6 nodes, the non-complex part of G is a forest of order 2. There are 2 forests. We have A140636(6) * 2, or 186 graphs.
If G has one complex component with 7 nodes, the non-complex part of G is one isolated vertex. We have A140636(7), or 809 graphs.
Finally if G is connected, we have A140636(8), or 11005 graphs.
The total is 3 + 12 + 2 + 4 + 39 + 13 + 186 + 809 + 11005 = 12073.

Svante Janson, Donald E. Knuth, Tomasz Łuczak and Boris Pittel, The Birth of the Giant Component.

Cf. A000088, A005195, A001429.
The labeled complement is A133686, covering A367869, connected A129271.
The complement is A134964, connected A005703.
The connected covering case is A140636.
The labeled version is A367867, covering A367868, connected A140638.
Set-systems not of this type are A367902, ranks A367906.
Set-systems of this type are A367903, ranks A367907.
For set-systems we have A368094, complement A368095.
For multiset partitions we have A368097, complement A368098.
Factorizations of this type are A368413, complement A368414.
Cf. A001349, A002494, A006125, A055621, A367769, A367770, A367901.

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}]],Select[Tuples[#], UnsameQ@@#&]=={}&]]],{n,0,5}] (* Gus Wiseman, Feb 14 2024 *)

a(n) = A000088(n) - A134964(n).

A368414 Number of factorizations of n into positive integers > 1 such that it is possible to choose a different prime factor of each factor.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 2, 1, 1, 3, 1, 3, 2, 2, 1, 4, 1, 2, 1, 3, 1, 5, 1, 1, 2, 2, 2, 5, 1, 2, 2, 4, 1, 5, 1, 3, 3, 2, 1, 5, 1, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 9, 1, 2, 3, 1, 2, 5, 1, 3, 2, 5, 1, 6, 1, 2, 3, 3, 2, 5, 1, 5, 1, 2, 1, 9, 2, 2, 2

Offset: 1

Gus Wiseman, Dec 29 2023

nonn

For example, the factorization f = 2*3*6 has two ways to choose a prime factor of each factor, namely (2,3,2) and (2,3,3), but neither of these has all different elements, so f is not counted under a(36).

			The a(n) factorizations for selected n:
  1    6      12     24      30       60        72      120
       2*3    2*6    2*12    2*15     2*30      2*36    2*60
              3*4    3*8     3*10     3*20      3*24    3*40
                     4*6     5*6      4*15      4*18    4*30
                             2*3*5    5*12      6*12    5*24
                                      6*10      8*9     6*20
                                      2*3*10            8*15
                                      2*5*6             10*12
                                      3*4*5             2*3*20
                                                        2*5*12
                                                        2*6*10
                                                        3*4*10
                                                        3*5*8
                                                        4*5*6

For labeled graphs: A133686, complement A367867, A367868, A140638.
For unlabeled graphs: A134964, complement A140637.
For set-systems: A367902, ranks A367906, complement A367903, ranks A367907.
For non-isomorphic set-systems: A368095, complement A368094, A368409.
Complementary non-isomorphic multiset partitions: A368097, A355529, A368411.
For non-isomorphic multiset partitions: A368098, A368100.
The complement is counted by A368413.
For non-isomorphic set multipartitions: A368422, complement A368421.
For divisors instead of prime factors: A370813, complement A370814.
A001055 counts factorizations, strict A045778.
A007716 counts non-isomorphic multiset partitions, connected A007718.
A058891 counts set-systems, unlabeled A000612, connected A323818.
A283877 counts non-isomorphic set-systems, connected A300913.
Cf. A340596, A340653, A367769, A367901, A368187, A368412, A368413.

facs[n_]:=If[n<=1,{{}},Join @@ Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]], {d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n], Select[Tuples[First/@FactorInteger[#]&/@#], UnsameQ@@#&]!={}&]],{n,100}]

a(n) = A001055(n) - A368413(n).

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

Original entry on oeis.org

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

A368094 Number of non-isomorphic set-systems of weight n contradicting a strict version of the axiom of choice.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 5, 12, 36, 97, 291

Offset: 0

Gus Wiseman, Dec 23 2023

A set-system is a finite set of finite nonempty sets. The weight of a set-system is the sum of cardinalities of its elements. Weight is generally not the same as number of vertices.

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.

			Non-isomorphic representatives of the a(5) = 1 through a(7) = 12 set-systems:
  {{1},{2},{3},{2,3}}  {{1},{2},{1,3},{2,3}}    {{1},{2},{1,2},{3,4,5}}
                       {{1},{2},{3},{1,2,3}}    {{1},{3},{2,3},{1,2,3}}
                       {{2},{3},{1,3},{2,3}}    {{1},{4},{1,4},{2,3,4}}
                       {{3},{4},{1,2},{3,4}}    {{2},{3},{2,3},{1,2,3}}
                       {{1},{2},{3},{4},{3,4}}  {{3},{1,2},{1,3},{2,3}}
                                                {{1},{2},{3},{1,3},{2,3}}
                                                {{1},{2},{3},{2,4},{3,4}}
                                                {{1},{2},{3},{4},{2,3,4}}
                                                {{1},{3},{4},{2,4},{3,4}}
                                                {{1},{4},{5},{2,3},{4,5}}
                                                {{2},{3},{4},{1,2},{3,4}}
                                                {{1},{2},{3},{4},{5},{4,5}}

Wikipedia, Axiom of choice.

The case of unlabeled graphs is A140637, complement A134964.
The case of labeled graphs is A367867, complement A133686.
The labeled version is A367903, ranks A367907.
The complement is counted by A368095, connected A368410.
Repeats allowed: A368097, ranks A355529, complement A368098, ranks A368100.
Minimal multiset partitions of this type are ranked by A368187.
The connected case is A368409.
Factorizations of this type are counted by A368413, complement A368414.
Allowing repeated edges gives A368421, complement A368422.
A000110 counts set partitions, non-isomorphic A000041.
A003465 counts covering set-systems, unlabeled A055621.
A007716 counts non-isomorphic multiset partitions, connected A007718.
A058891 counts set-systems, unlabeled A000612, connected A323818.
A283877 counts non-isomorphic set-systems, connected A300913.
Cf. A302545, A306005, A316983, A317533, A319616, A321155, A321405, A326031, A330223, A330227, A367905.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]] /@ Cases[Subsets[set],{i,_}];
mpm[n_]:=Join@@Table[Union[Sort[Sort/@(#/.x_Integer:>s[[x]])]& /@ sps[Range[n]]], {s,Flatten[MapIndexed[Table[#2,{#1}]&,#]]& /@ IntegerPartitions[n]}];
brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])], {p,Permutations[Union@@m]}]]];
Table[Length[Union[brute/@Select[mpm[n], UnsameQ@@#&&And@@UnsameQ@@@# && Select[Tuples[#], UnsameQ@@#&]=={}&]]],{n,0,8}]

A370593 Number of integer partitions of n such that it is not possible to choose a different prime factor of each part.

Original entry on oeis.org

0, 1, 1, 2, 4, 5, 10, 12, 19, 26, 38, 51, 71, 94, 126, 165, 219, 285, 369, 472, 605, 766, 973, 1226, 1538, 1917, 2387, 2955, 3657, 4497, 5532, 6754, 8251, 10033, 12190, 14748, 17831, 21471, 25825, 30976, 37111, 44331, 52897, 62952, 74829, 88755, 105145, 124307

Offset: 0

Gus Wiseman, Feb 29 2024

nonn

			The a(0) = 0 through a(7) = 12 partitions:
  .  (1)  (11)  (21)   (22)    (41)     (33)      (61)
                (111)  (31)    (221)    (42)      (322)
                       (211)   (311)    (51)      (331)
                       (1111)  (2111)   (222)     (421)
                               (11111)  (321)     (511)
                                        (411)     (2221)
                                        (2211)    (3211)
                                        (3111)    (4111)
                                        (21111)   (22111)
                                        (111111)  (31111)
                                                  (211111)
                                                  (1111111)

The complement for divisors instead of factors is A239312, ranks A368110.
These partitions have ranks A355529, complement A368100.
The complement for set-systems is A367902, ranks A367906, unlabeled A368095.
The version for set-systems is A367903, ranks A367907, unlabeled A368094.
For unlabeled multiset partitions we have A368097, complement A368098.
The version for factorizations is A368413, complement A368414.
The complement is counted by A370592.
For a unique choice we have A370594, ranks A370647.
A006530 gives greatest prime factor, least A020639.
A027746 lists prime factors, A112798 indices, length A001222.
A355741 counts choices of a prime factor of each prime index.
Cf. A000040, A000720, A133686, A355739, A355740, A367771, A367867, A367905, A370583, A370585, A370586, A370636.

Table[Length[Select[IntegerPartitions[n], Length[Select[Tuples[If[#==1,{},First/@FactorInteger[#]]&/@#], UnsameQ@@#&]]==0&]],{n,0,30}]

a(n) = A000041(n) - A370592(n).

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

A368095 Number of non-isomorphic set-systems of weight n satisfying a strict version of the axiom of choice.

Original entry on oeis.org

1, 1, 2, 4, 8, 17, 39, 86, 208, 508, 1304

Offset: 0

Gus Wiseman, Dec 24 2023

A set-system is a finite set of finite nonempty sets. The weight of a set-system is the sum of cardinalities of its elements.

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.

			Non-isomorphic representatives of the a(1) = 1 through a(5) = 17 set-systems:
  {1}  {12}    {123}      {1234}        {12345}
       {1}{2}  {1}{23}    {1}{234}      {1}{2345}
               {2}{12}    {12}{34}      {12}{345}
               {1}{2}{3}  {13}{23}      {14}{234}
                          {3}{123}      {23}{123}
                          {1}{2}{34}    {4}{1234}
                          {1}{3}{23}    {1}{2}{345}
                          {1}{2}{3}{4}  {1}{23}{45}
                                        {1}{24}{34}
                                        {1}{4}{234}
                                        {2}{13}{23}
                                        {2}{3}{123}
                                        {3}{13}{23}
                                        {4}{12}{34}
                                        {1}{2}{3}{45}
                                        {1}{2}{4}{34}
                                        {1}{2}{3}{4}{5}

For labeled graphs we have A133686, complement A367867.
For unlabeled graphs we have A134964, complement A140637.
For set-systems we have A367902, complement A367903.
These set-systems have BII-numbers A367906, complement A367907.
The complement is A368094, connected A368409.
Repeats allowed: A368098, ranks A368100, complement A368097, ranks A355529.
Minimal multiset partitions not of this type are counted by A368187.
The connected case is A368410.
Factorizations of this type are counted by A368414, complement A368413.
Allowing repeated edges gives A368422, complement A368421.
A000110 counts set-partitions, non-isomorphic A000041.
A003465 counts covering set-systems, unlabeled A055621.
A007716 counts non-isomorphic multiset partitions, connected A007718.
A058891 counts set-systems, unlabeled A000612, connected A323818.
A283877 counts non-isomorphic set-systems, connected A300913.
Cf. A001055, A007717, A302545, A306005, A316983, A319560, A321194, A321405, A330223, A367769, A367901.

Table[Length[Select[bmp[n], UnsameQ@@#&&And@@UnsameQ@@@#&&Select[Tuples[#], UnsameQ@@#&]!={}&]], {n,0,10}]

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

A368098 Number of non-isomorphic multiset partitions of weight n satisfying a strict version of the axiom of choice.

Original entry on oeis.org

1, 1, 3, 7, 21, 54, 165, 477, 1501, 4736, 15652

Offset: 0

Gus Wiseman, Dec 25 2023

A multiset partition is a finite multiset of finite nonempty multisets. The weight of a multiset partition is the sum of cardinalities of its elements. Weight is generally not the same as number of vertices.

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.

			Non-isomorphic representatives of the a(1) = 1 through a(4) = 21 multiset partitions:
  {{1}}  {{1,1}}    {{1,1,1}}      {{1,1,1,1}}
         {{1,2}}    {{1,2,2}}      {{1,1,2,2}}
         {{1},{2}}  {{1,2,3}}      {{1,2,2,2}}
                    {{1},{2,2}}    {{1,2,3,3}}
                    {{1},{2,3}}    {{1,2,3,4}}
                    {{2},{1,2}}    {{1},{1,2,2}}
                    {{1},{2},{3}}  {{1,1},{2,2}}
                                   {{1,2},{1,2}}
                                   {{1},{2,2,2}}
                                   {{1,2},{2,2}}
                                   {{1},{2,3,3}}
                                   {{1,2},{3,3}}
                                   {{1},{2,3,4}}
                                   {{1,2},{3,4}}
                                   {{1,3},{2,3}}
                                   {{2},{1,2,2}}
                                   {{3},{1,2,3}}
                                   {{1},{2},{3,3}}
                                   {{1},{2},{3,4}}
                                   {{1},{3},{2,3}}
                                   {{1},{2},{3},{4}}

Wikipedia, Axiom of choice.

The case of labeled graphs is A133686, complement A367867.
The case of unlabeled graphs is A134964, complement A140637 (apparently).
Set-systems of this type are A367902, ranks A367906, connected A368410.
The complimentary set-systems are A367903, ranks A367907, connected A368409.
For set-systems we have A368095, complement A368094.
The complement is A368097, ranks A355529.
These multiset partitions have ranks A368100.
The connected case is A368412, complement A368411.
Factorizations of this type are counted by A368414, complement A368413.
For set multipartitions we have A368422, complement A368421.
A000110 counts set partitions, non-isomorphic A000041.
A003465 counts covering set-systems, unlabeled A055621.
A007716 counts non-isomorphic multiset partitions, connected A007718.
A058891 counts set-systems, unlabeled A000612, connected A323818.
A283877 counts non-isomorphic set-systems, connected A300913.
Cf. A302545, A306005, A316983, A317533, A319616, A330223, A330227, A368187.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]] /@ Cases[Subsets[set],{i,_}];
mpm[n_]:=Join@@Table[Union[Sort[Sort/@(#/.x_Integer:>s[[x]])]& /@ sps[Range[n]]], {s,Flatten[MapIndexed[Table[#2,{#1}]&,#]]& /@ IntegerPartitions[n]}];
brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])], {p,Permutations[Union@@m]}]]];
Table[Length[Union[brute/@Select[mpm[n], Select[Tuples[#],UnsameQ@@#&]!={}&]]], {n,0,6}]

cp's OEIS Frontend

A140637 Number of unlabeled graphs of positive excess with n nodes.

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A368414 Number of factorizations of n into positive integers > 1 such that it is possible to choose a different prime factor of each factor.

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

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A368094 Number of non-isomorphic set-systems of weight n contradicting a strict version of the axiom of choice.

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A370593 Number of integer partitions of n such that it is not possible to choose a different prime factor of each part.

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

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

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