A367867 -id:A367867 - cp's OEIS frontend

A368730 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, such that it is not possible to choose a different element from each.

0, 0, 0, 0, 6, 180, 4560, 117600, 3234588, 96119982, 3092585310, 107542211535, 4029055302855, 162040513972623, 6970457656110039, 319598974394563500, 15568332397812799920, 803271954062642638830, 43778508937914677872788, 2513783434620146896920843

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(4) = 6 set-systems:
  {{1},{2},{1,2},{3,4}}
  {{1},{3},{1,3},{2,4}}
  {{1},{4},{1,4},{2,3}}
  {{2},{3},{1,4},{2,3}}
  {{2},{4},{1,3},{2,4}}
  {{3},{4},{1,2},{3,4}}

Andrew Howroyd, Table of n, a(n) for n = 0..200
Eric Weisstein's World of Mathematics, Graph Loop.
ProofWiki, Definition:Loop-Graph

The case of a unique choice appears to be A000272.
The version without the choice condition is A368597, non-covering A014068.
The complement appears to be A333331.
The non-covering case is A368596, allowing edges of any size A368600.
Allowing any number of edges of any size gives A367903, ranks A367907.
Allowing any number of non-singletons gives A367868, non-covering A367867.
A000085 counts set partitions into singletons or pairs.
A006125 counts graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
A100861 counts set partitions into singletons or pairs by number of pairs.
A111924 counts set partitions into singletons or pairs by length.
A322661 counts labeled covering half-loop-graphs, connected A062740.
Cf. A000666, A116508, A133686, A367863, A367869, A367901, A368532.

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

a(n) = A368596(n) + A368597(n) - A014068(n). - Andrew Howroyd, Jan 10 2024

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

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).

Original entry on oeis.org

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

A368187 Divisor-minimal numbers whose prime indices of prime indices contradict a strict version of the axiom of choice.

Original entry on oeis.org

2, 9, 21, 25, 49, 57, 115, 121, 133, 159, 195, 289, 361, 371, 393, 455, 507, 515, 529, 555, 845, 897, 915, 917, 933, 957, 961, 1007, 1067, 1183, 1235, 1295, 1335, 1443, 1681, 2093, 2095, 2135, 2157, 2177, 2193, 2197, 2233, 2265, 2343, 2369, 2379, 2405, 2489

Offset: 1

Gus Wiseman, Dec 29 2023

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

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 terms together with their prime indices begin:
     2: {1}
     9: {2,2}
    21: {2,4}
    25: {3,3}
    49: {4,4}
    57: {2,8}
   115: {3,9}
   121: {5,5}
   133: {4,8}
   159: {2,16}
   195: {2,3,6}
   289: {7,7}
   361: {8,8}
   371: {4,16}
   393: {2,32}
   455: {3,4,6}

Wikipedia, Axiom of choice.

The version for BII-numbers of set-systems is A368532.
A000110 counts set partitions, non-isomorphic A000041.
A003465 counts covering set-systems, unlabeled A055621.
A007716 counts non-isomorphic multiset partitions, connected A007718.
Cf. A134964, A140637, A355529, A367905, A367907.
Cf. A367867, A367903, A368094, A368097, A368413, A368421.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
vmin[y_]:=Select[y,Function[s, Select[DeleteCases[y,s], Divisible[s,#]&]=={}]];
Select[Range[100],Select[Tuples[prix /@ prix[#]],UnsameQ@@#&]=={}&]//vmin

A370639 Number of subsets of {1..n} containing n such that it is possible to choose a different binary index of each element.

Original entry on oeis.org

0, 1, 2, 3, 7, 10, 15, 22, 61, 81, 112, 154, 207, 276, 355, 464, 1771, 2166, 2724, 3445, 4246, 5292, 6420, 7922, 9586, 11667, 13768, 16606, 19095, 22825, 26498, 31421, 187223, 213684, 247670, 289181, 331301, 385079, 440411, 510124, 575266, 662625, 747521

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(6) = 15 subsets:
  .  {1}  {2}    {3}    {4}      {5}      {6}
          {1,2}  {1,3}  {1,4}    {1,5}    {1,6}
                 {2,3}  {2,4}    {2,5}    {2,6}
                        {3,4}    {3,5}    {3,6}
                        {1,2,4}  {4,5}    {4,6}
                        {1,3,4}  {1,2,5}  {5,6}
                        {2,3,4}  {1,3,5}  {1,2,6}
                                 {2,3,5}  {1,3,6}
                                 {2,4,5}  {1,4,6}
                                 {3,4,5}  {1,5,6}
                                          {2,3,6}
                                          {2,5,6}
                                          {3,4,6}
                                          {3,5,6}
                                          {4,5,6}

Wikipedia, Axiom of choice.

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 of this type are A368414/A370814, complement A368413/A370813.
For prime instead of binary indices we have A370586, differences of A370582.
The complement for prime indices is A370587, differences of A370583.
The complement is counted by A370589, differences of A370637.
Partial sums are A370636.
The complement has partial sums A370637/A370643, minima A370642/A370644.
The case of a unique choice is A370641, differences of A370638.
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, A367770, A367772, A367905, A367909, A367912, A368094, A368095, A368109, A370640.

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

First differences of A370636.

a(19)-a(42) from Alois P. Heinz, Mar 09 2024

A368422 Number of non-isomorphic set multipartitions of weight n satisfying a strict version of the axiom of choice.

Original entry on oeis.org

1, 1, 2, 4, 9, 18, 43, 95, 233, 569

Offset: 0

Gus Wiseman, Dec 26 2023

A set multipartition is a finite multiset of finite nonempty sets. The weight of a set multipartition 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 sequence of nonempty sets, it is possible to choose a sequence containing an element from each. In the strict version, the elements of this sequence must be distinct, meaning none is chosen more than once.

			Non-isomorphic representatives of the a(1) = 1 through a(5) = 18 set multipartitions:
  {{1}}  {{1,2}}    {{1,2,3}}      {{1,2,3,4}}        {{1,2,3,4,5}}
         {{1},{2}}  {{1},{2,3}}    {{1,2},{1,2}}      {{1},{2,3,4,5}}
                    {{2},{1,2}}    {{1},{2,3,4}}      {{1,2},{3,4,5}}
                    {{1},{2},{3}}  {{1,2},{3,4}}      {{1,4},{2,3,4}}
                                   {{1,3},{2,3}}      {{2,3},{1,2,3}}
                                   {{3},{1,2,3}}      {{4},{1,2,3,4}}
                                   {{1},{2},{3,4}}    {{1},{2,3},{2,3}}
                                   {{1},{3},{2,3}}    {{1},{2},{3,4,5}}
                                   {{1},{2},{3},{4}}  {{1},{2,3},{4,5}}
                                                      {{1},{2,4},{3,4}}
                                                      {{1},{4},{2,3,4}}
                                                      {{2},{1,3},{2,3}}
                                                      {{2},{3},{1,2,3}}
                                                      {{3},{1,3},{2,3}}
                                                      {{4},{1,2},{3,4}}
                                                      {{1},{2},{3},{4,5}}
                                                      {{1},{2},{4},{3,4}}
                                                      {{1},{2},{3},{4},{5}}

Wikipedia, Axiom of choice.

The case of unlabeled graphs is A134964, complement A140637.
Set multipartitions have ranks A302478, cf. A073576.
The case of labeled graphs is A133686, complement A367867.
The complement without repeats is A368094 connected A368409.
Without repeats we have A368095, connected A368410.
The complement allowing repeats is A368097, ranks A355529.
Allowing repeated elements gives A368098, ranks A368100.
Factorizations of this type are counted by A368414, complement A368413.
The complement is counted by 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, A317533, A318360, A367903.

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],And@@UnsameQ@@@#&&Select[Tuples[#], UnsameQ@@#&]!={}&]]],{n,0,6}]

A369143 Number of labeled simple graphs with n edges and 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, 0, 30, 1335, 47460, 1651230, 59636640, 2284113762, 93498908580, 4099070635935, 192365988161490, 9646654985111430, 515736895712230192, 29321225548502776980, 1768139644819077541440, 112805126206185257070660, 7595507651522103787077270, 538504704005397535690160274

Offset: 0

Gus Wiseman, Jan 21 2024

nonn

			The term a(5) = 30 counts all permutations of the graph {{1,2},{1,3},{1,4},{2,3},{2,4}}.

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

The version without the choice condition is A116508, covering A367863.
The complement is A137916.
Allowing any number of edges gives A367867, covering A367868.
The version with loops is A368596, covering A368730, unlabeled A368835.
For set-systems we have A368600, for any number of edges A367903.
The covering case is A369144.
A006125 counts simple graphs, unlabeled A000088.
A058891 counts set-systems (without singletons A016031), unlabeled A000612.
Cf. A000085, A057500, A062740, A129271, A133686, A333331, A367769, A367869, A367901, A367907.

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

a(n) = A116508(n) - A137916(n). - Andrew Howroyd, Feb 02 2024

a(8) onwards from Andrew Howroyd, Feb 02 2024

A370589 Number of subsets of {1..n} containing 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, 1, 6, 17, 42, 67, 175, 400, 870, 1841, 3820, 7837, 15920, 30997, 63370, 128348, 258699, 520042, 1043284, 2090732, 4186382, 8379022, 16765549, 33540664, 67092258, 134198633, 268412631, 536844414, 1073710403, 2147296425, 4294753612, 8589686922, 17179580003

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 binary indices of {1,4,5} are {{1},{3},{1,3}}, from which it is not possible to choose three different elements, so S is counted under a(3).
The binary indices of S = {1,6,8,9} are {{1},{2,3},{4},{1,4}}, from which it is not possible to choose four different elements, so S is counted under a(9).
The a(0) = 0 through a(6) = 17 subsets:
  .  .  .  {1,2,3}  {1,2,3,4}  {1,4,5}      {2,4,6}
                               {1,2,3,5}    {1,2,3,6}
                               {1,2,4,5}    {1,2,4,6}
                               {1,3,4,5}    {1,2,5,6}
                               {2,3,4,5}    {1,3,4,6}
                               {1,2,3,4,5}  {1,3,5,6}
                                            {1,4,5,6}
                                            {2,3,4,6}
                                            {2,3,5,6}
                                            {2,4,5,6}
                                            {3,4,5,6}
                                            {1,2,3,4,6}
                                            {1,2,3,5,6}
                                            {1,2,4,5,6}
                                            {1,3,4,5,6}
                                            {2,3,4,5,6}
                                            {1,2,3,4,5,6}

Wikipedia, Axiom of choice.

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 A370586, differences of A370582.
For prime indices we have A370587, differences of A370583.
Partial sums are A370637/A370643, minima A370642/A370644.
The complement is counted by A370639, partial sums A370636.
The version for a unique choice is A370641, partial sums A370638.
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, A326702, A355739, A355740, A367772, A367905, A367909, A367912, A368094, A368095, A368109, A370640.

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

a(19)-a(35) from Alois P. Heinz, Mar 09 2024

A368410 Number of non-isomorphic connected set-systems of weight n satisfying a strict version of the axiom of choice.

Original entry on oeis.org

0, 1, 1, 2, 3, 7, 15, 32, 80, 198, 528

Offset: 0

Gus Wiseman, Dec 25 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(1) = 1 through a(6) = 15 set-systems:
  {1}  {12}  {123}    {1234}    {12345}      {123456}
             {2}{12}  {13}{23}  {14}{234}    {125}{345}
                      {3}{123}  {23}{123}    {134}{234}
                                {4}{1234}    {15}{2345}
                                {2}{13}{23}  {34}{1234}
                                {2}{3}{123}  {5}{12345}
                                {3}{13}{23}  {1}{14}{234}
                                             {12}{13}{23}
                                             {1}{23}{123}
                                             {13}{24}{34}
                                             {14}{24}{34}
                                             {3}{14}{234}
                                             {3}{23}{123}
                                             {3}{4}{1234}
                                             {4}{14}{234}

Wikipedia, Axiom of choice.

For unlabeled graphs we have A005703, connected case of A134964.
For labeled graphs we have A129271, connected case of A133686.
The complement for labeled graphs is A140638, connected case of A367867.
The complement without connectedness is A367903, ranks A367907.
Without connectedness we have A368095, ranks A367906,
Complement with repeats: A368097, connected case of A368411, ranks A355529.
The complement is counted by A368409, connected case of A368094.
With repeats allowed: A368412, connected case of A368098, ranks A368100.
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, A140637, A302545, A306005, A321194, A321405, A367902, A368414, A368422.

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]}]]];
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[Union[brute/@Select[mpm[n], UnsameQ@@#&&And@@UnsameQ@@@#&&Length[csm[#]]==1&&Select[Tuples[#], UnsameQ@@#&]!={}&]]],{n,0,6}]

A368421 Number of non-isomorphic set multipartitions of weight n contradicting a strict version of the axiom of choice.

Original entry on oeis.org

0, 0, 1, 2, 7, 16, 47, 116, 325, 861

Offset: 0

Gus Wiseman, Dec 26 2023

A set multipartition is a finite multiset of finite nonempty sets. The weight of a set multipartition 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 sequence of nonempty sets Y, it is possible to choose a sequence containing an element from each. In the strict version, the elements of this sequence must be distinct, meaning none is chosen more than once.

			Non-isomorphic representatives of the a(2) = 1 through a(5) = 16 set multipartitions:
  {{1},{1}}  {{1},{1},{1}}  {{1},{1},{2,3}}    {{1},{1},{2,3,4}}
             {{1},{2},{2}}  {{1},{2},{1,2}}    {{2},{1,2},{1,2}}
                            {{2},{2},{1,2}}    {{3},{3},{1,2,3}}
                            {{1},{1},{1},{1}}  {{1},{1},{1},{2,3}}
                            {{1},{1},{2},{2}}  {{1},{1},{3},{2,3}}
                            {{1},{2},{2},{2}}  {{1},{2},{2},{1,2}}
                            {{1},{2},{3},{3}}  {{1},{2},{2},{3,4}}
                                               {{1},{2},{3},{2,3}}
                                               {{1},{3},{3},{2,3}}
                                               {{2},{2},{2},{1,2}}
                                               {{1},{1},{1},{1},{1}}
                                               {{1},{1},{2},{2},{2}}
                                               {{1},{2},{2},{2},{2}}
                                               {{1},{2},{2},{3},{3}}
                                               {{1},{2},{3},{3},{3}}
                                               {{1},{2},{3},{4},{4}}

Wikipedia, Axiom of choice.

The case of unlabeled graphs is A140637, complement A134964.
Set multipartitions have ranks A302478, cf. A073576.
The case of labeled graphs is A367867, complement A133686.
With distinct edges we have A368094 connected A368409.
The complement with distinct edges is A368095, connected A368410.
Allowing repeated elements gives A368097, ranks A355529.
The complement allowing repeats is A368098, ranks A368100.
Factorizations of this type are counted by A368413, complement A368414.
The complement is counted by 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, A318360, A367903, A367905, A367907.

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],And@@UnsameQ@@@#&&Select[Tuples[#], UnsameQ@@#&]=={}&]]],{n,0,6}]

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.

cp's OEIS Frontend

A368730 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, such that it is not possible to choose a different element from each.

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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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A368187 Divisor-minimal numbers whose prime indices of prime indices contradict a strict version of the axiom of choice.

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A370639 Number of subsets of {1..n} containing n such that it is possible to choose a different binary index of each element.

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A368422 Number of non-isomorphic set multipartitions of weight n satisfying a strict version of the axiom of choice.

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A369143 Number of labeled simple graphs with n edges and n vertices such that it is not possible to choose a different vertex from each edge (non-choosable).

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A370589 Number of subsets of {1..n} containing n such that it is not possible to choose a different binary index of each element.

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A368410 Number of non-isomorphic connected set-systems of weight n satisfying a strict version of the axiom of choice.