A319559 -id:A319559 - cp's OEIS frontend

A245567 Number of antichain covers of a labeled n-set such that for every two distinct elements in the n-set, there is a set in the antichain cover containing one of the elements but not the other.

2, 1, 1, 5, 76, 5993, 7689745, 2414465044600, 56130437141763247212112, 286386577668298408602599478477358234902247

Offset: 0

Patrick De Causmaecker, Jul 25 2014

This is the number of antichain covers such that the induced partition contains only singletons. The induced partition of {{1,2},{2,3},{1,3},{3,4}} is {{1},{2},{3},{4}}, while the induced partition of {{1,2,3},{2,3,4}} is {{1},{2,3},{4}}.
This sequence is related to A006126. See 1st formula.
The sequence is also related to Dedekind numbers through Stirling numbers of the second kind. See 2nd formula.
Sets of subsets of the described type are said to be T_0. - Gus Wiseman, Aug 14 2019

			For n = 0, a(0) = 2 by the antisets {}, {{}}.
For n = 1, a(1) = 1 by the antiset {{1}}.
For n = 2, a(2) = 1 by the antiset {{1},{2}}.
For n = 3, a(3) = 5 by the antisets {{1},{2},{3}}, {{1,2},{1,3}}, {{1,2},{2,3}}, {{1,3},{2,3}}, {{1,2},{1,3},{2,3}}.

Patrick De Causmaecker and Stefan De Wannemacker, On the number of antichains of sets in a finite universe, arXiv:1407.4288 [math.CO], 2014.

Cf. A000372 (Dedekind numbers), A006126 (Number of antichain covers of a labeled n-set).
Sequences counting and ranking T_0 structures:
  A000112 (unlabeled topologies),
  A001035 (topologies),
  A059201 (covering set-systems),
  A245567 (antichain covers),
  A309615 (covering set-systems closed under intersection),
  A316978 (factorizations),
  A319559 (unlabeled set-systems by weight),
  A319564 (integer partitions),
  A319637 (unlabeled covering set-systems),
  A326939 (covering sets of subsets),
  A326940 (set-systems),
  A326941 (sets of subsets),
  A326943 (covering sets of subsets closed under intersection),
  A326944 (covering sets of subsets with {} and closed under intersection),
  A326945 (sets of subsets closed under intersection),
  A326946 (unlabeled set-systems),
  A326947 (BII-numbers of set-systems),
  A326948 (connected set-systems),
  A326949 (unlabeled sets of subsets),
  A326950 (antichains),
  A326959 (set-systems closed under intersection),
  A327013 (unlabeled covering set-systems closed under intersection),
  A327016 (BII-numbers of topologies).

dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Length[Select[Subsets[Subsets[Range[n]]],Union@@#==Range[n]&&stableQ[#,SubsetQ]&&UnsameQ@@dual[#]&]],{n,0,3}] (* Gus Wiseman, Aug 14 2019 *)

A000372(n) = Sum_{k=0..n} S(n+1,k+1)*a(k).
a(n) = A006126(n) - Sum_{k=1..n-1} S(n,k)*a(k).
Were n > 0 and S(n,k) is the number of ways to partition a set of n elements into k nonempty subsets.
Inverse binomial transform of A326950, if we assume a(0) = 1. - Gus Wiseman, Aug 14 2019

Definition corrected by Patrick De Causmaecker, Oct 10 2014

a(9), based on A000372, from Patrick De Causmaecker, Jun 01 2023

A330055 Number of non-isomorphic set-systems of weight n with no singletons or endpoints.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 1, 3, 5, 16, 24, 90, 179, 567, 1475, 4623, 13650, 44475, 144110, 492017, 1706956, 6124330, 22442687, 84406276, 324298231, 1273955153, 5106977701, 20885538133, 87046940269, 369534837538, 1596793560371, 7019424870960, 31374394197536, 142514998263015

Offset: 0

Gus Wiseman, Nov 30 2019

nonn

A set-system is a finite set of finite nonempty set of positive integers. A singleton is an edge of size 1. An endpoint is a vertex appearing only once (degree 1). The weight of a set-system is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(7) = 1 through a(10) = 16 set-systems:
  {12}{13}{123}  {12}{134}{234}    {12}{134}{1234}    {12}{1345}{2345}
                 {12}{34}{1234}    {123}{124}{134}    {123}{124}{1234}
                 {12}{13}{24}{34}  {12}{13}{14}{234}  {123}{145}{2345}
                                   {12}{13}{23}{123}  {12}{345}{12345}
                                   {12}{13}{24}{134}  {12}{13}{124}{134}
                                                      {12}{13}{124}{234}
                                                      {12}{13}{14}{1234}
                                                      {12}{13}{24}{1234}
                                                      {12}{13}{245}{345}
                                                      {12}{13}{45}{2345}
                                                      {12}{34}{123}{124}
                                                      {12}{34}{125}{345}
                                                      {12}{34}{135}{245}
                                                      {13}{24}{123}{124}
                                                      {12}{13}{14}{23}{24}
                                                      {12}{13}{24}{35}{45}

Andrew Howroyd, Table of n, a(n) for n = 0..50
Wikipedia, Degree (graph theory)

The labeled version is A330056.
The "multi" version is A320665.
Non-isomorphic set-systems with no singletons are A306005.
Non-isomorphic set-systems with no endpoints are A330054.
Non-isomorphic set-systems counted by vertices are A000612.
Non-isomorphic set-systems counted by weight are A283877.
Cf. A007716, A055621, A302545, A317533, A317794, A319559, A321405, A330052, A330058.

WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)}
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
K(q, t, k)={my(g=x*Ser(WeighT(Vec(sum(j=1, #q, my(g=gcd(t, q[j])); g*x^(q[j]/g)) + O(x*x^k), -k)))); (1-x)*g-subst(g,x,x^2)}
S(q, t, k)={(x-x^2)*sum(j=1, #q, if(t%q[j]==0, q[j])) + O(x*x^k)}
a(n)={if(n==0, 1, my(s=0); forpart(q=n, s+=permcount(q)*polcoef(exp(sum(t=1, n, subst(K(q, t, n\t)-S(q,t,n\t),x,x^t)/t )), n)); s/n!)} \\ Andrew Howroyd, Jan 27 2024

a(11) onwards from Andrew Howroyd, Jan 27 2024

A330102 BII-number of the VDD-normalization of the set-system with BII-number n.

Original entry on oeis.org

0, 1, 1, 3, 4, 5, 5, 7, 1, 3, 3, 11, 33, 19, 19, 15, 4, 5, 33, 19, 20, 21, 37, 23, 5, 7, 19, 15, 37, 23, 51, 31, 4, 33, 5, 19, 20, 37, 21, 23, 5, 19, 7, 15, 37, 51, 23, 31, 20, 37, 37, 51, 52, 53, 53, 55, 21, 23, 23, 31, 53, 55, 55, 63, 64, 65, 65, 67, 68, 69, 69

Offset: 0

Gus Wiseman, Dec 04 2019

nonn

First differs from A330101 at a(148) = 274, A330101(148) = 545, with corresponding set-systems 274: {{2},{1,3},{1,4}} and 545: {{1},{2,3},{2,4}}.
A set-system is a finite set of finite nonempty sets of positive integers.
We define the VDD (vertex-degrees decreasing) normalization of a set-system 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 sets is first by length and then lexicographically.
For example, 156 is the BII-number of {{3},{4},{1,2},{1,3}}, which has the following normalizations, together with their BII-numbers:
  Brute-force:    2067: {{1},{2},{1,3},{3,4}}
  Lexicographic:   165: {{1},{4},{1,2},{2,3}}
  VDD:             525: {{1},{3},{1,2},{2,4}}
  MM:              270: {{2},{3},{1,2},{1,4}}
  BII:             150: {{2},{4},{1,2},{1,3}}
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. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every set-system has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, the BII-number of {{2},{1,3}} is 18. Elements of a set-system are sometimes called edges.

			56 is the BII-number of {{3},{1,3},{2,3}}, which has VDD-normalization {{1},{1,2},{1,3}} with BII-number 21, so a(56) = 21.

Wikipedia, Idempotence

This sequence is idempotent and its image/fixed points are A330100.
Non-isomorphic multiset partitions are A007716.
Unlabeled spanning set-systems counted by vertices are A055621.
Unlabeled set-systems counted by weight are A283877.
Cf. A000120, A000612, A048793, A070939, A300913, A319559, A321405, A326031, A326754, A330061, A330101.
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).

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
fbi[q_]:=If[q=={},0,Total[2^q]/2];
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[fbi[fbi/@sysnorm[bpe/@bpe[n]]],{n,0,100}]

A326971 Number of unlabeled set-systems on n vertices whose dual is a weak antichain.

Original entry on oeis.org

1, 2, 5, 24, 1267

Offset: 0

Gus Wiseman, Aug 10 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) = 24 set-systems:
  {}  {}     {}               {}
      {{1}}  {{1}}            {{1}}
             {{1,2}}          {{1,2}}
             {{1},{2}}        {{1},{2}}
             {{1},{2},{1,2}}  {{1,2,3}}
                              {{1},{2,3}}
                              {{1},{2},{3}}
                              {{1},{2},{1,2}}
                              {{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 set-systems are A000612.
Unlabeled set-systems whose dual is strict are A326946.
The labeled version is A326968.
The version with empty edges allowed is A326969.
The T_0 case (with strict dual) is A326972.
The covering case is A326973 (first differences).
Cf. A319559, A326951, A326965, A326966, A326970, A326974, A326975, A326978.

A330052 Number of non-isomorphic set-systems of weight n with at least one endpoint.

Original entry on oeis.org

0, 1, 2, 4, 8, 18, 40, 94, 228, 579, 1508, 4092, 11478, 33337, 100016, 309916, 990008, 3257196, 11021851, 38314009, 136657181, 499570867, 1869792499, 7158070137, 28003286261, 111857491266, 455852284867, 1893959499405, 8017007560487, 34552315237016, 151534813272661

Offset: 0

Gus Wiseman, Nov 30 2019

nonn

A set-system is a finite set of finite nonempty sets of positive integers. An endpoint is a vertex appearing only once (degree 1). The weight of a set-system is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(1) = 1 through a(5) = 18 multiset partitions:
  {1}  {12}    {123}      {1234}        {12345}
       {1}{2}  {1}{12}    {1}{123}      {1}{1234}
               {1}{23}    {12}{13}      {12}{123}
               {1}{2}{3}  {1}{234}      {12}{134}
                          {12}{34}      {1}{2345}
                          {1}{2}{13}    {12}{345}
                          {1}{2}{34}    {1}{12}{13}
                          {1}{2}{3}{4}  {1}{12}{23}
                                        {1}{12}{34}
                                        {1}{2}{123}
                                        {1}{2}{134}
                                        {1}{2}{345}
                                        {1}{23}{45}
                                        {2}{13}{14}
                                        {1}{2}{3}{12}
                                        {1}{2}{3}{14}
                                        {1}{2}{3}{45}
                                        {1}{2}{3}{4}{5}

Andrew Howroyd, Table of n, a(n) for n = 0..50
Wikipedia, Degree (graph theory)

The complement is counted by A330054.
The multiset partition version is A330058.
Non-isomorphic set-systems with at least one singleton are A330053.
Non-isomorphic set-systems counted by vertices are A000612.
Non-isomorphic set-systems counted by weight are A283877.
Cf. A007716, A055621, A306005, A317533, A317794, A319559, A320665, A330055, A330056.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
brute[{}]:={};brute[m_]:=If[Union@@m!={}&&Union@@m!=Range[Max@@Flatten[m]],brute[m/.Rule@@@Table[{(Union@@m)[[i]],i},{i,Length[Union@@m]}]],First[Sort[brute[m,1]]]];brute[m_,1]:=Table[Sort[Sort/@(m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])],{p,Permutations[Union@@m]}];
Table[Length[Select[Union[brute/@Join@@mps/@strnorm[n]],UnsameQ@@#&&And@@UnsameQ@@@#&&Min@@Length/@Split[Sort[Join@@#]]==1&]],{n,0,5}]

a(n) = A283877(n) - A330054(n). - Andrew Howroyd, Jan 27 2024

a(11) onwards from Andrew Howroyd, Jan 27 2024

A330054 Number of non-isomorphic set-systems of weight n with no endpoints.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 4, 4, 16, 26, 87, 181, 570, 1453, 4464, 13038, 41548, 132217, 442603, 1506803, 5305174, 19092816, 70548770, 266495254, 1029835424, 4063610148, 16366919221, 67217627966, 281326631801, 1199048810660, 5201341196693, 22950740113039, 102957953031700

Offset: 0

Gus Wiseman, Nov 30 2019

nonn

A set-system is a finite set of finite nonempty set of positive integers. An endpoint is a vertex appearing only once (degree 1). The weight of a set-system is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(0) = 1 through a(8) = 16 multiset partitions (empty columns not shown):
  0  {1}{2}{12}  {12}{13}{23}    {13}{23}{123}      {12}{134}{234}
                 {1}{23}{123}    {1}{3}{23}{123}    {1}{234}{1234}
                 {1}{2}{13}{23}  {3}{12}{13}{23}    {12}{34}{1234}
                 {1}{2}{3}{123}  {1}{2}{3}{13}{23}  {1}{12}{34}{234}
                                                    {12}{13}{24}{34}
                                                    {1}{2}{134}{234}
                                                    {1}{2}{34}{1234}
                                                    {2}{13}{14}{234}
                                                    {2}{13}{23}{123}
                                                    {3}{13}{23}{123}
                                                    {1}{2}{13}{24}{34}
                                                    {1}{2}{3}{14}{234}
                                                    {1}{2}{3}{23}{123}
                                                    {1}{2}{3}{4}{1234}
                                                    {2}{3}{12}{13}{23}
                                                    {1}{2}{3}{4}{12}{34}

Andrew Howroyd, Table of n, a(n) for n = 0..50
Wikipedia, Degree (graph theory)

The complement is counted by A330052.
The multiset partition version is A302545.
Non-isomorphic set-systems with no singletons are A306005.
Non-isomorphic set-systems counted by vertices are A000612.
Non-isomorphic set-systems counted by weight are A283877.
Cf. A007716, A055621, A317533, A317794, A319559, A320665, A330055, A330056, A330058.

WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)}
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
K(q, t, k)={my(g=1+x*Ser(WeighT(Vec(sum(j=1, #q, my(g=gcd(t, q[j])); g*x^(q[j]/g)) + O(x*x^k), -k)))); (1-x)*g - subst(g,x,x^2)}
a(n)={if(n==0, 1, my(s=0); forpart(q=n, s+=permcount(q)*polcoef(exp(sum(t=1, n, subst(K(q,t,n\t)/t,x,x^t) )), n)); s/n!)} \\ Andrew Howroyd, Jan 27 2024

a(11) onwards from Andrew Howroyd, Jan 27 2024

A319566 Number of non-isomorphic connected T_0 set systems of weight n.

Original entry on oeis.org

1, 1, 0, 1, 2, 3, 8, 17, 41, 103, 276

Offset: 0

Gus Wiseman, Sep 23 2018

In a set system, two vertices are equivalent if in every block the presence of the first is equivalent to the presence of the second. The T_0 condition means that there are no equivalent vertices.

The weight of a set system is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(1) = 1 through a(6) = 8 set systems:
1:        {{1}}
3:     {{2},{1,2}}
4:    {{1,3},{2,3}}
     {{1},{2},{1,2}}
5:  {{2},{3},{1,2,3}}
    {{2},{1,3},{2,3}}
    {{3},{1,3},{2,3}}
6: {{3},{1,4},{2,3,4}}
   {{3},{2,3},{1,2,3}}
   {{1,2},{1,3},{2,3}}
   {{1,3},{2,4},{3,4}}
   {{1,4},{2,4},{3,4}}
  {{1},{2},{3},{1,2,3}}
  {{1},{2},{1,3},{2,3}}
  {{2},{3},{1,3},{2,3}}

Cf. A007716, A007718, A049311, A056156, A059201, A283877, A316980.

Cf. A319557, A319558, A319559, A319560, A319564, A319565, A319567.

A321446 Number of (0,1)-matrices with n ones, no zero rows or columns, and distinct rows and columns.

Original entry on oeis.org

1, 1, 2, 10, 72, 624, 6522, 80178, 1129368, 17917032, 316108752, 6138887616, 130120838400, 2989026225696, 73964789192400, 1961487062520720, 55495429438186920, 1668498596700706440, 53122020640948010640, 1785467619718933936560, 63175132023953553400440

Offset: 0

Gus Wiseman, Nov 13 2018

nonn

			The a(3) = 10 matrices:
  [1 1] [1 1] [1 0] [0 1]
  [1 0] [0 1] [1 1] [1 1]
.
  [1 0 0] [1 0 0] [0 1 0] [0 1 0] [0 0 1] [0 0 1]
  [0 1 0] [0 0 1] [1 0 0] [0 0 1] [1 0 0] [0 1 0]
  [0 0 1] [0 1 0] [0 0 1] [1 0 0] [0 1 0] [1 0 0]

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

Cf. A000612, A007716, A049311, A101370, A120733, A135589, A283877, A316980, A319559, A321515, A321586, A321587, A321588, A369285.

prs2mat[prs_]:=Table[Count[prs,{i,j}],{i,Union[First/@prs]},{j,Union[Last/@prs]}];
Table[Length[Select[Subsets[Tuples[Range[n],2],{n}],And[Union[First/@#]==Range[Max@@First/@#],Union[Last/@#]==Range[Max@@Last/@#],UnsameQ@@prs2mat[#],UnsameQ@@Transpose[prs2mat[#]]]&]],{n,6}]

\\ Q(m, n, wf) defined in A321588.
seq(n)={my(R=vectorv(n,m,Q(m,n,w->1 + y^w + O(y*y^n)))); for(i=2, #R, R[i] -= i*R[i-1]); Vec(1 + vecsum(vecsum(R)))} \\ Andrew Howroyd, Jan 24 2024

a(7) onwards from Andrew Howroyd, Jan 20 2024

A326949 Number of unlabeled T_0 sets of subsets of {1..n}.

Original entry on oeis.org

2, 4, 10, 68, 3838, 37320356

Offset: 0

Gus Wiseman, Aug 08 2019

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}}. The T_0 condition means that the dual is strict (no repeated edges).

			Non-isomorphic representatives of the a(0) = 2 through a(2) = 10 sets of sets:
  {}    {}        {}
  {{}}  {{}}      {{}}
        {{1}}     {{1}}
        {{},{1}}  {{},{1}}
                  {{1},{2}}
                  {{2},{1,2}}
                  {{},{1},{2}}
                  {{},{2},{1,2}}
                  {{1},{2},{1,2}}
                  {{},{1},{2},{1,2}}

The non-T_0 version is A003180.
The labeled version is A326941.
The covering case is A326942 (first differences).
The case without empty edges is A326946.
Cf. A000371, A000612, A003181, A059052, A245567, A316978, A319559, A319564, A319637, A326939, A326940.

a(n) = 2 * A326946(n).

a(5) from Max Alekseyev, Oct 11 2023

A326950 Number of T_0 antichains of nonempty subsets of {1..n}.

Original entry on oeis.org

1, 2, 4, 12, 107, 6439, 7726965, 2414519001532, 56130437161079183223017, 286386577668298409107773412840148848120595

Offset: 0

Gus Wiseman, Aug 08 2019

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}}. The T_0 condition means that the dual is strict (no repeated edges).

			The a(0) = 1 through a(3) = 12 antichains:
  {}  {}     {}         {}
      {{1}}  {{1}}      {{1}}
             {{2}}      {{2}}
             {{1},{2}}  {{3}}
                        {{1},{2}}
                        {{1},{3}}
                        {{2},{3}}
                        {{1,2},{1,3}}
                        {{1,2},{2,3}}
                        {{1},{2},{3}}
                        {{1,3},{2,3}}
                        {{1,2},{1,3},{2,3}}

Antichains of nonempty sets are A014466.
T_0 set-systems are A326940.
The covering case is A245567.
Cf. A006126, A059201, A059052, A245567, A319559, A319564, A326030, A326946, A326947.

dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],stableQ[#,SubsetQ]&&UnsameQ@@dual[#]&]],{n,0,3}]

Binomial transform of A245567, if we assume A245567(0) = 1.

a(5)-a(8) from Andrew Howroyd, Aug 14 2019

a(9), based on A245567, from Patrick De Causmaecker, Jun 01 2023

cp's OEIS Frontend

A245567 Number of antichain covers of a labeled n-set such that for every two distinct elements in the n-set, there is a set in the antichain cover containing one of the elements but not the other.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A330055 Number of non-isomorphic set-systems of weight n with no singletons or endpoints.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Extensions

A330102 BII-number of the VDD-normalization of the set-system with BII-number n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A326971 Number of unlabeled set-systems on n vertices whose dual is a weak antichain.

Views

Author

Keywords

Comments

Examples

Crossrefs

A330052 Number of non-isomorphic set-systems of weight n with at least one endpoint.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A330054 Number of non-isomorphic set-systems of weight n with no endpoints.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Extensions

A319566 Number of non-isomorphic connected T_0 set systems of weight n.

Views

Author

Keywords

Comments

Examples

Crossrefs

A321446 Number of (0,1)-matrices with n ones, no zero rows or columns, and distinct rows and columns.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica