A000612 -id:A000612 - cp's OEIS frontend

A108798 Number of nonisomorphic systems enumerated by A102894; that is, the number of inequivalent closure operators in which the empty set is closed. Also, the number of union-closed sets with n elements that contain the universe and the empty set.

1, 1, 3, 14, 165, 14480, 108281182, 2796163091470050

Offset: 0

Don Knuth, Jul 01 2005

Also the number of unlabeled finite sets of subsets of {1..n} that contain {} and {1..n} and are closed under intersection. - Gus Wiseman, Aug 02 2019

			From _Gus Wiseman_, Aug 02 2019: (Start)
Non-isomorphic representatives of the a(0) = 1 through a(3) = 14 union-closed sets of sets:
  {}  {}{1}  {}{12}        {}{123}
             {}{2}{12}     {}{3}{123}
             {}{1}{2}{12}  {}{23}{123}
                           {}{1}{23}{123}
                           {}{3}{23}{123}
                           {}{13}{23}{123}
                           {}{2}{3}{23}{123}
                           {}{2}{13}{23}{123}
                           {}{3}{13}{23}{123}
                           {}{12}{13}{23}{123}
                           {}{2}{3}{13}{23}{123}
                           {}{3}{12}{13}{23}{123}
                           {}{2}{3}{12}{13}{23}{123}
                           {}{1}{2}{3}{12}{13}{23}{123}
(End)

Maria Paola Bonacina and Nachum Dershowitz, Canonical ground Horn theories, Lecture Notes in Computer Science 7797, 35-71 (2013).
G. Brinkmann and R. Deklerck, Generation of Union-Closed Sets and Moore Families, Journal of Integer Sequences, Vol.21 (2018), Article 18.1.7.
G. Brinkmann and R. Deklerck, Generation of Union-Closed Sets and Moore Families, arXiv:1701.03751 [math.CO], 2017.
Christopher S. Flippen, Minimal Sets, Union-Closed Families, and Frankl's Conjecture, Master's thesis, Virginia Commonwealth Univ., 2023.

The labeled version is A102894.

Cf. A000612, A001930, A003180, A102895, A102897, A108800, A193674, A193675, A326867, A326869, A326883.

a(n) = A108800(n)/2.

a(6) added (using A193674) by N. J. A. Sloane, Aug 02 2011

Added a(7), and reference to union-closed sets. - Gunnar Brinkmann, Feb 05 2018

A326972 Number of unlabeled set-systems on n vertices whose dual is a (strict) antichain, also called unlabeled T_1 set-systems.

Original entry on oeis.org

1, 2, 4, 20, 1232

Offset: 0

Gus Wiseman, Aug 11 2019

A set-system is a finite set of finite nonempty sets. The dual of a set-system has, for each vertex, one edge consisting of the indices (or positions) of the edges containing that vertex. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. An antichain is a set of sets, none of which is a subset of any other.

			Non-isomorphic representatives of the a(0) = 1 through a(3) = 20 set-systems:
  {}  {}     {}               {}
      {{1}}  {{1}}            {{1}}
             {{1},{2}}        {{1},{2}}
             {{1},{2},{1,2}}  {{1},{2},{3}}
                              {{1},{2},{1,2}}
                              {{1,2},{1,3},{2,3}}
                              {{1},{2},{3},{2,3}}
                              {{1},{2},{1,3},{2,3}}
                              {{1},{2},{3},{1,2,3}}
                              {{3},{1,2},{1,3},{2,3}}
                              {{1},{2},{3},{1,3},{2,3}}
                              {{1,2},{1,3},{2,3},{1,2,3}}
                              {{1},{2},{3},{2,3},{1,2,3}}
                              {{2},{3},{1,2},{1,3},{2,3}}
                              {{1},{2},{1,3},{2,3},{1,2,3}}
                              {{1},{2},{3},{1,2},{1,3},{2,3}}
                              {{3},{1,2},{1,3},{2,3},{1,2,3}}
                              {{1},{2},{3},{1,3},{2,3},{1,2,3}}
                              {{2},{3},{1,2},{1,3},{2,3},{1,2,3}}
                              {{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}

Unlabeled set-systems are A000612.
Unlabeled set-systems whose dual is strict are A326946.
The version with empty edges allowed is A326951.
The labeled version is A326965.
The version where the dual is not required to be strict is A326971.
The covering version is A326974 (first differences).
Cf. A059523, A319559, A319637, A326973, A326976, A326977, A326979.

A367916 Number of sets of nonempty subsets of {1..n} with the same number of edges as covered vertices.

Original entry on oeis.org

1, 2, 6, 45, 1376, 161587, 64552473, 85987037645, 386933032425826, 6005080379837219319, 328011924848834642962619, 64153024576968812343635391868, 45547297603829979923254392040011994, 118654043008142499115765307533395739785599

Offset: 0

Gus Wiseman, Dec 08 2023

nonn

			The a(0) = 1 through a(2) = 6 set-systems:
  {}  {}     {}
      {{1}}  {{1}}
             {{2}}
             {{1},{2}}
             {{1},{1,2}}
             {{2},{1,2}}

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

The covering case is A054780.
For graphs we have A367862, covering A367863, unlabeled A006649.
These set-systems have ranks A367917.
A000372 counts antichains, covering A006126, nonempty A014466.
A003465 counts set-systems covering {1..n}, unlabeled A055621.
A058891 counts set-systems, unlabeled A000612.
A059201 counts covering T_0 set-systems.
A136556 counts set-systems on {1..n} with n edges.
Cf. A092918, A102896, A133686, A306445, A323818, A355740, A367770, A367869, A367901, A367902, A367905.

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

\\ Here b(n) is A054780(n).
b(n) = sum(k=0, n, (-1)^(n-k) * binomial(n,k) * binomial(2^k-1, n))
a(n) = sum(k=0, n, binomial(n,k) * b(k)) \\ Andrew Howroyd, Dec 29 2023

Binomial transform of A054780.

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

Gus Wiseman, Jan 05 2024

nonn

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

			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}}

Andrew Howroyd, Table of n, a(n) for n = 0..50
Eric Weisstein's World of Mathematics, Graph Loop.

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.

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}]

a(n) = polcoef(G(n, O(x*x^n)), n) \\ G defined in A070166. - Andrew Howroyd, Jan 09 2024

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

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

A368835 Number of unlabeled n-edge loop-graphs with at most n vertices such that it is not possible to choose a different vertex from each edge.

Original entry on oeis.org

0, 0, 0, 1, 5, 23, 98, 394, 1560, 6181, 24655, 99701, 410513, 1725725, 7423757, 32729320, 148027044, 687188969, 3275077017, 16022239940, 80431483586, 414094461610, 2185052929580, 11808696690600, 65312048149993, 369408792148714, 2135111662435080, 12601466371445619

Offset: 0

Gus Wiseman, Jan 13 2024

nonn

			Non-isomorphic representatives of the a(4) = 5 loop-graphs:
  {{1,1},{2,2},{3,3},{1,2}}
  {{1,1},{2,2},{1,2},{1,3}}
  {{1,1},{2,2},{1,2},{3,4}}
  {{1,1},{2,2},{1,3},{2,3}}
  {{1,1},{1,2},{1,3},{2,3}}

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

The case of a unique choice is A000081, row sums of A106234.
The labeled version is A368596, covering A368730.
Without the choice condition we have A368598.
The complement is A368984, row sums of A368926.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A006125 counts graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
A014068 counts loop-graphs, unlabeled A000666.
A058891 counts set-systems (without singletons A016031), unlabeled A000612.
A322661 counts labeled covering half-loop-graphs, connected A062740.
Cf. A116508, A129271, A133686, A137916, A137917, A333331, A367863, A368836, A368924, A368927.

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

a(n) = A368598(n) - A368984(n). - Andrew Howroyd, Jan 14 2024

a(8) onwards from Andrew Howroyd, Jan 14 2024

A317795 Number of non-isomorphic set-systems spanning n vertices with no singletons.

Original entry on oeis.org

1, 0, 1, 6, 172, 611852, 200253853704512, 263735716028826427334553305221120, 5609038300883759793482640992086670066496449147691597380832361377955840

Offset: 0

Gus Wiseman, Aug 07 2018

nonn

			Non-isomorphic representatives of the a(3) = 6 set-systems:
  {123}
  {12}{13}
  {12}{123}
  {12}{13}{23}
  {12}{13}{123}
  {12}{13}{23}{123}

First differences of A317794.

Cf. A000088, A000612, A003180, A007716, A055621, A283877, A300913, A306005, A317533, A317757, A319876.

sysnorm[{}]:={};sysnorm[m_]:=If[Union@@m!=Range[Max@@Flatten[m]],sysnorm[m/.Rule@@@Table[{(Union@@m)[[i]],i},{i,Length[Union@@m]}]],First[Sort[sysnorm[m,1]]]];sysnorm[m_,aft_]:=If[Length[Union@@m]<=aft,{m},With[{mx=Table[Count[m,i,{2}],{i,Select[Union@@m,#>=aft&]}]},Union@@(sysnorm[#,aft+1]&/@Union[Table[Map[Sort,m/.{par+aft-1->aft,aft->par+aft-1},{0,1}],{par,First/@Position[mx,Max[mx]]}]])]];
Table[Length[Union[sysnorm/@Select[Subsets[Select[Subsets[Range[n]],Length[#]>1&]],Union@@#==Range[n]&]]],{n,4}]

More terms from Gus Wiseman, Dec 13 2018

A330061 MM-number of the VDD-normalization of the multiset of multisets with MM-number n.

Original entry on oeis.org

1, 2, 3, 4, 3, 6, 7, 8, 9, 6, 3, 12, 13, 14, 15, 16, 3, 18, 19, 12, 21, 6, 7, 24, 9, 26, 27, 28, 13, 30, 3, 32, 15, 6, 35, 36, 37, 38, 39, 24, 3, 42, 13, 12, 45, 14, 13, 48, 49, 18, 15, 52, 53, 54, 15, 56, 57, 26, 3, 60, 37, 6, 63, 64, 39, 30, 3, 12, 35, 70

Offset: 1

Gus Wiseman, Dec 03 2019

nonn

We define the VDD (vertex-degrees decreasing) normalization of a multiset of multisets to be obtained by first normalizing so that the vertices cover an initial interval of positive integers, then applying all permutations to the vertex set, then selecting only the representatives whose vertex-degrees are weakly decreasing, and finally taking the least of these representatives, where the ordering of multisets is first by length and then lexicographically.
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 multiset of multisets with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.
For example, 15301 is the MM-number of {{3},{1,2},{1,1,4}}, which has the following normalizations together with their MM-numbers:
  Brute-force:    43287: {{1},{2,3},{2,2,4}}
  Lexicographic:  43143: {{1},{2,4},{2,2,3}}
  VDD:            15515: {{2},{1,3},{1,1,4}}
  MM:             15265: {{2},{1,4},{1,1,3}}

Wikipedia, Idempotence

This sequence is idempotent and its image/fixed points are A330060.
Non-isomorphic multiset partitions are A007716.
MM-weight is A302242.
Cf. A000612, A055621, A056239, A112798, A283877, A316983, A317533, A320456, A330098, A330102, A330103, A330105.
Other fixed points:
- Brute-force: A330104 (multisets of multisets), A330107 (multiset partitions), A330099 (set-systems).
- Lexicographic: A330120 (multisets of multisets), A330121 (multiset partitions), A330110 (set-systems).
- VDD: A330060 (multisets of multisets), A330097 (multiset partitions), A330100 (set-systems).
- MM: A330108 (multisets of multisets), A330122 (multiset partitions), A330123 (set-systems).
- BII: A330109 (set-systems).

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
sysnorm[m_]:=If[Union@@m!={}&&Union@@m!=Range[Max@@Flatten[m]],sysnorm[m/.Rule@@@Table[{(Union@@m)[[i]],i},{i,Length[Union@@m]}]],First[Sort[sysnorm[m,1]]]];
sysnorm[m_,aft_]:=If[Length[Union@@m]<=aft,{m},With[{mx=Table[Count[m,i,{2}],{i,Select[Union@@m,#>=aft&]}]},Union@@(sysnorm[#,aft+1]&/@Union[Table[Map[Sort,m/.{par+aft-1->aft,aft->par+aft-1},{0,1}],{par,First/@Position[mx,Max[mx]]}]])]];
Table[Map[Times@@Prime/@#&,sysnorm[primeMS/@primeMS[n]],{0,1}],{n,100}]

A330229 Number of fully chiral set-systems covering n vertices.

Original entry on oeis.org

1, 1, 2, 42, 21336

Offset: 0

Gus Wiseman, Dec 08 2019

A set-system is a finite set of finite nonempty sets. It is fully chiral if every permutation of the vertices gives a different representative.

			The a(3) = 42 set-systems:
  {1}{2}{13}    {1}{2}{12}{13}    {1}{2}{12}{13}{123}
  {1}{2}{23}    {1}{2}{12}{23}    {1}{2}{12}{23}{123}
  {1}{3}{12}    {1}{3}{12}{13}    {1}{3}{12}{13}{123}
  {1}{3}{23}    {1}{3}{13}{23}    {1}{3}{13}{23}{123}
  {2}{3}{12}    {2}{3}{12}{23}    {2}{3}{12}{23}{123}
  {2}{3}{13}    {2}{3}{13}{23}    {2}{3}{13}{23}{123}
  {1}{12}{23}   {1}{2}{13}{123}
  {1}{13}{23}   {1}{2}{23}{123}
  {2}{12}{13}   {1}{3}{12}{123}
  {2}{13}{23}   {1}{3}{23}{123}
  {3}{12}{13}   {2}{3}{12}{123}
  {3}{12}{23}   {2}{3}{13}{123}
  {1}{12}{123}  {1}{12}{23}{123}
  {1}{13}{123}  {1}{13}{23}{123}
  {2}{12}{123}  {2}{12}{13}{123}
  {2}{23}{123}  {2}{13}{23}{123}
  {3}{13}{123}  {3}{12}{13}{123}
  {3}{23}{123}  {3}{12}{23}{123}

The non-covering version is A330282.
Costrict (or T_0) covering set-systems are A059201.
BII-numbers of fully chiral set-systems are A330226.
Non-isomorphic, fully chiral multiset partitions are A330227.
Fully chiral partitions are counted by A330228.
Fully chiral covering set-systems are A330229.
Fully chiral factorizations are A330235.
MM-numbers of fully chiral multisets of multisets are A330236.
Cf. A000612, A055621, A083323, A283877, A319559, A319564, A330224, A330234.

graprms[m_]:=Union[Table[Sort[Sort/@(m/.Rule@@@Table[{p[[i]],i},{i,Length[p]}])],{p,Permutations[Union@@m]}]];
Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],Union@@#==Range[n]&&Length[graprms[#]]==n!&]],{n,0,3}]

Binomial transform is A330282.

A368599 Number of non-isomorphic n-element sets of singletons or pairs of elements of {1..n} with union {1..n}, or unlabeled loop-graphs with n edges covering n vertices.

Original entry on oeis.org

1, 1, 2, 5, 13, 34, 97, 277, 825, 2486, 7643, 23772, 74989, 238933, 769488, 2500758, 8199828, 27106647, 90316944, 303182461, 1025139840, 3490606305, 11967066094, 41302863014, 143493606215, 501772078429, 1765928732426, 6254738346969, 22294413256484, 79968425399831

Offset: 0

Gus Wiseman, Jan 06 2024

nonn

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

			The a(0) = 1 through a(4) = 13 set-systems:
  {}  {{1}}  {{1},{2}}    {{1},{2},{3}}        {{1},{2},{3},{4}}
             {{1},{1,2}}  {{1},{2},{1,3}}      {{1},{2},{3},{1,4}}
                          {{1},{1,2},{1,3}}    {{1},{2},{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},{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}}

Andrew Howroyd, Table of n, a(n) for n = 0..50
Eric Weisstein's World of Mathematics, Graph Loop.

For any number of edges we have A000666, A054921, A322700.
For any number of edges of any size we have A055621, non-covering A000612.
For edges of any size we have A368186, covering case of A368731.
The labeled version is A368597, covering case of A014068.
This is the covering case of A368598.
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. A007716, A007717, A058891, A070166, A122848, A283877, A302545, A320663, A322661, A339741, A339887, A339888.

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

a(n) = polcoef(G(n, O(x*x^n)) - if(n, G(n-1, O(x*x^n))), n) \\ G defined in A070166. - Andrew Howroyd, Jan 09 2024

a(n) = A070166(n,n) - A070166(n-1,n) for n > 0. - Andrew Howroyd, Jan 09 2024

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

A326973 Number of unlabeled set-systems covering n vertices whose dual is a weak antichain.

Original entry on oeis.org

1, 1, 3, 19, 1243

Offset: 0

Gus Wiseman, Aug 11 2019

A set-system is a finite set of finite nonempty sets. The dual of a set-system has, for each vertex, one edge consisting of the indices (or positions) of the edges containing that vertex. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. A weak antichain is a multiset of sets, none of which is a proper subset of any other.

			Non-isomorphic representatives of the a(0) = 1 through a(3) = 19 set-systems:
  {}  {{1}}  {{1,2}}          {{1,2,3}}
             {{1},{2}}        {{1},{2,3}}
             {{1},{2},{1,2}}  {{1},{2},{3}}
                              {{1,2},{1,3},{2,3}}
                              {{1},{2,3},{1,2,3}}
                              {{1},{2},{3},{2,3}}
                              {{1},{2},{1,3},{2,3}}
                              {{1},{2},{3},{1,2,3}}
                              {{3},{1,2},{1,3},{2,3}}
                              {{1},{2},{3},{1,3},{2,3}}
                              {{1,2},{1,3},{2,3},{1,2,3}}
                              {{1},{2},{3},{2,3},{1,2,3}}
                              {{2},{3},{1,2},{1,3},{2,3}}
                              {{1},{2},{1,3},{2,3},{1,2,3}}
                              {{1},{2},{3},{1,2},{1,3},{2,3}}
                              {{3},{1,2},{1,3},{2,3},{1,2,3}}
                              {{1},{2},{3},{1,3},{2,3},{1,2,3}}
                              {{2},{3},{1,2},{1,3},{2,3},{1,2,3}}
                              {{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}

Unlabeled covering set-systems are A055621.
The labeled version is A326970.
The non-covering case is A326971 (partial sums).
The case that is also T_0 is the T_1 case A326974.
Cf. A000612, A059523, A319637, A326966, A326968, A326972, A326975, A326978.

cp's OEIS Frontend

A108798 Number of nonisomorphic systems enumerated by A102894; that is, the number of inequivalent closure operators in which the empty set is closed. Also, the number of union-closed sets with n elements that contain the universe and the empty set.

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A326972 Number of unlabeled set-systems on n vertices whose dual is a (strict) antichain, also called unlabeled T_1 set-systems.

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A367916 Number of sets of nonempty subsets of {1..n} with the same number of edges as covered vertices.

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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.

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Mathematica

PARI

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A368835 Number of unlabeled n-edge loop-graphs with at most n vertices such that it is not possible to choose a different vertex from each edge.

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Mathematica

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A317795 Number of non-isomorphic set-systems spanning n vertices with no singletons.

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Mathematica

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A330061 MM-number of the VDD-normalization of the multiset of multisets with MM-number n.

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Mathematica

A330229 Number of fully chiral set-systems covering n vertices.

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Mathematica