A327062 -id:A327062 - cp's OEIS frontend

A327352 Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of antichains of nonempty subsets of {1..n} with spanning edge-connectivity k.

1, 1, 1, 4, 1, 14, 4, 1, 83, 59, 23, 2, 1232, 2551, 2792, 887, 107, 10, 1

Offset: 0

Gus Wiseman, Sep 10 2019

An antichain is a set of sets, none of which is a subset of any other.

The spanning edge-connectivity of a set-system is the minimum number of edges that must be removed (without removing incident vertices) to obtain a set-system that is disconnected or covers fewer vertices.

			Triangle begins:
     1
     1    1
     4    1
    14    4    1
    83   59   23    2
  1232 2551 2792  887  107   10    1
Row n = 3 counts the following antichains:
  {}             {{1,2,3}}      {{1,2},{1,3},{2,3}}
  {{1}}          {{1,2},{1,3}}
  {{2}}          {{1,2},{2,3}}
  {{3}}          {{1,3},{2,3}}
  {{1,2}}
  {{1,3}}
  {{2,3}}
  {{1},{2}}
  {{1},{3}}
  {{2},{3}}
  {{1},{2,3}}
  {{2},{1,3}}
  {{3},{1,2}}
  {{1},{2},{3}}

Row sums are A014466.
Column k = 0 is A327355.
The unlabeled version is A327438.
Cf. A052446, A327062, A327071, A327103, A327111, A327144, A327351, A327353.

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]]]]]]]]];
stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
spanEdgeConn[vts_,eds_]:=Length[eds]-Max@@Length/@Select[Subsets[eds],Union@@#!=vts||Length[csm[#]]!=1&];
Table[Length[Select[stableSets[Subsets[Range[n],{1,n}],SubsetQ],spanEdgeConn[Range[n],#]==k&]],{n,0,4},{k,0,2^n}]//.{foe___,0}:>{foe}

A327060 Number of non-isomorphic weight-n weak antichains of multisets where every two vertices appear together in some edge (cointersecting).

Original entry on oeis.org

1, 1, 3, 4, 9, 11, 30, 42, 103, 194, 443

Offset: 0

Gus Wiseman, Aug 18 2019

A multiset partition is a finite multiset of finite nonempty multisets. It is a weak antichain if no part is a proper submultiset of any other.

			Non-isomorphic representatives of the a(0) = 1 through a(5) = 11 multiset partitions:
  {}  {{1}}  {{11}}    {{111}}      {{1111}}        {{11111}}
             {{12}}    {{122}}      {{1122}}        {{11222}}
             {{1}{1}}  {{123}}      {{1222}}        {{12222}}
                       {{1}{1}{1}}  {{1233}}        {{12233}}
                                    {{1234}}        {{12333}}
                                    {{11}{11}}      {{12344}}
                                    {{12}{12}}      {{12345}}
                                    {{12}{22}}      {{11}{122}}
                                    {{1}{1}{1}{1}}  {{12}{222}}
                                                    {{33}{123}}
                                                    {{1}{1}{1}{1}{1}}

Antichains are A000372.
The BII-numbers of these set-systems are the intersection of A326853 and A326704.
Cointersecting set-systems are A327039.
The set-system version is A327057, with covering case A327058.
Cf. A006126, A007716, A051185, A305844, A326965, A327020, A327059, A327062.

A327356 Number of connected separable antichains of nonempty sets covering n vertices (vertex-connectivity 1).

Original entry on oeis.org

0, 0, 1, 3, 40, 1365

Offset: 0

Gus Wiseman, Sep 11 2019

An antichain is a set of sets, none of which is a subset of any other. It is covering if there are no isolated vertices.

The vertex-connectivity of a set-system is the minimum number of vertices that must be removed (along with any resulting empty edges) to obtain a non-connected set-system or singleton. Note that this means a single node has vertex-connectivity 0.

			Non-isomorphic representatives of the a(4) = 40 set-systems:
  {{1,2},{1,3,4}}
  {{1,2},{1,3},{1,4}}
  {{1,2},{1,3},{2,4}}
  {{1,2},{1,3},{1,4},{2,3}}

Column k = 1 of A327351.
The graphical case is A327336.
The unlabeled version is A327436.
Cf. A014466, A048143, A307249, A326786, A326950, A327062, A327112, A327114, A327334.

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]]]]]]]]];
stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
vertConnSys[vts_,eds_]:=Min@@Length/@Select[Subsets[vts],Function[del,Length[del]==Length[vts]-1||csm[DeleteCases[DeleteCases[eds,Alternatives@@del,{2}],{}]]!={Complement[vts,del]}]];
Table[Length[Select[stableSets[Subsets[Range[n],{1,n}],SubsetQ],vertConnSys[Range[n],#]==1&]],{n,0,4}]

A327806 Triangle read by rows where T(n,k) is the number of antichains of sets with n vertices and vertex-connectivity >= k.

Original entry on oeis.org

1, 2, 0, 5, 1, 0, 19, 5, 2, 0, 167, 84, 44, 17, 0

Offset: 0

Gus Wiseman, Sep 26 2019

An antichain is a set of nonempty sets, none of which is a subset of any other.

The vertex-connectivity of a set-system is the minimum number of vertices that must be removed (along with any resulting empty edges) to obtain a non-connected set-system or singleton. Note that this means a single node has vertex-connectivity 0.

			Triangle begins:
    1
    2   0
    5   1   0
   19   5   2   0
  167  84  44  17   0

Except for the first column, same as the covering case A327350.
Column k = 0 is A014466 (antichains).
Column k = 1 is A048143 (clutters), if we assume A048143(0) = A048143(1) = 0.
Column k = 2 is A275307 (blobs), if we assume A275307(1) = A275307(2) = 0.
The unlabeled version is A327807.
The case for vertex connectivity exactly k is A327351.
Cf. A014466, A293606, A326704, A327057, A327062, A327125, A327358.

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]]]]]]]]];
stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
vertConnSys[vts_,eds_]:=Min@@Length/@Select[Subsets[vts],Function[del,Length[del]==Length[vts]-1||csm[DeleteCases[DeleteCases[eds,Alternatives@@del,{2}],{}]]!={Complement[vts,del]}]];
Table[Length[Select[stableSets[Subsets[Range[n],{1,n}],SubsetQ],vertConnSys[Range[n],#]>=k&]],{n,0,4},{k,0,n}]

A327059 Number of pairwise intersecting set-systems covering a subset of {1..n} whose dual is a weak antichain.

Original entry on oeis.org

1, 2, 4, 10, 178

Offset: 0

Gus Wiseman, Aug 18 2019

A set-system is a finite set of finite nonempty sets. Its elements are sometimes called edges. 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.

			The a(0) = 1 through a(3) = 10 set-systems:
  {}  {}     {}      {}
      {{1}}  {{1}}   {{1}}
             {{2}}   {{2}}
             {{12}}  {{3}}
                     {{12}}
                     {{13}}
                     {{23}}
                     {{123}}
                     {{12}{13}{23}}
                     {{12}{13}{23}{123}}

Intersecting set-systems are A051185.
The BII-numbers of these set-systems are the intersection of A326910 and A326966.
Set-systems whose dual is a weak antichain are A326968.
The covering version is A327058.
The unlabeled multiset partition version is A327060.
Cf. A000372, A305844, A326951, A326965, A326970, A327057, A327062.

dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Length[Select[stableSets[Subsets[Range[n],{1,n}],Intersection[#1,#2]=={}&],stableQ[dual[#],SubsetQ]&]],{n,0,3}]

Binomial transform of A327058.

A327424 Number of unlabeled, non-connected or empty antichains of nonempty subsets of {1..n}.

Original entry on oeis.org

1, 1, 2, 4, 10, 33, 234, 16579

Offset: 0

Gus Wiseman, Sep 26 2019

An antichain is a set of nonempty sets, none of which is a subset of any other. A singleton is considered to be connected.

			Non-isomorphic representatives of the a(0) = 1 through a(4) = 10 antichains:
  {}  {}  {}         {}             {}
          {{1},{2}}  {{1},{2}}      {{1},{2}}
                     {{1},{2,3}}    {{1},{2,3}}
                     {{1},{2},{3}}  {{1},{2},{3}}
                                    {{1},{2,3,4}}
                                    {{1,2},{3,4}}
                                    {{1},{2},{3,4}}
                                    {{1},{2},{3},{4}}
                                    {{1},{2,4},{3,4}}
                                    {{1},{2,3},{2,4},{3,4}}

Partial sums of the positive-index terms of A327426.
The covering case is A327426.
The labeled version is A327354.
The labeled covering case is A120338.
Unlabeled antichains that are either not connected or not covering are A327437.
The case without empty antichains is A327808.
Cf. A014466, A293606, A326704, A327062, A327355.

A327438 Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of unlabeled antichains of nonempty subsets of {1..n} with spanning edge-connectivity k.

Original entry on oeis.org

1, 1, 1, 3, 1, 6, 2, 1, 15, 7, 5, 2, 52, 53, 62, 31, 9, 1, 1

Offset: 0

Gus Wiseman, Sep 11 2019

An antichain is a set of sets, none of which is a subset of any other.

The spanning edge-connectivity of a set-system is the minimum number of edges that must be removed (without removing incident vertices) to obtain a set-system that is disconnected or covers fewer vertices.

			Triangle begins:
   1
   1  1
   3  1
   6  2  1
  15  7  5  2
  52 53 62 31  9  1  1
The antichains counted in row n = 4 are the following:
  0             {1234}         {12}{134}{234}     {123}{124}{134}{234}
  {1}           {12}{134}      {123}{124}{134}    {12}{13}{14}{23}{24}{34}
  {12}          {123}{124}     {12}{13}{24}{34}
  {123}         {12}{13}{14}   {12}{13}{14}{234}
  {1}{2}        {12}{13}{24}   {12}{13}{14}{23}{24}
  {1}{23}       {12}{13}{234}
  {12}{13}      {12}{13}{14}{23}
  {1}{234}
  {12}{34}
  {1}{2}{3}
  {1}{2}{34}
  {2}{13}{14}
  {12}{13}{23}
  {1}{2}{3}{4}
  {4}{12}{13}{23}

Row sums are A306505.
Column k = 0 is A327437.
The labeled version is A327352.
Cf. A014466, A052446, A327062, A327071, A327103, A327111, A327144, A327352, A327353.

A327808 Number of unlabeled, disconnected, nonempty antichains of subsets of {1..n}.

Original entry on oeis.org

0, 0, 1, 3, 9, 32, 233, 16578

Offset: 0

Gus Wiseman, Sep 26 2019

An antichain is a set of nonempty sets, none of which is a subset of any other. A singleton is considered to be connected.

			Non-isomorphic representatives of the a(2) = 1 through a(4) = 9 antichains:
   {{1},{2}}  {{1},{2}}      {{1},{2}}
              {{1},{2,3}}    {{1},{2,3}}
              {{1},{2},{3}}  {{1},{2},{3}}
                             {{1},{2,3,4}}
                             {{1,2},{3,4}}
                             {{1},{2},{3,4}}
                             {{1},{2},{3},{4}}
                             {{2},{1,3},{1,4}}
                             {{4},{1,2},{1,3},{2,3}}

The labeled version is A327354 - 1.
The covering case is A327426.
Unlabeled antichains that are either not connected or not covering are A327437.
The version with empty antichains allowed is A327424.
Cf. A000372, A014466, A120338, A293606, A326704, A327062, A327355.

a(n) = A327424(n) - 1.

A327807 Triangle read by rows where T(n,k) is the number of unlabeled antichains of sets with n vertices and vertex-connectivity >= k.

Original entry on oeis.org

1, 2, 0, 4, 1, 0, 9, 3, 2, 0, 29, 14, 10, 6, 0, 209, 157, 128, 91, 54, 0

Offset: 0

Gus Wiseman, Sep 26 2019

An antichain is a set of sets, none of which is a subset of any other.

The vertex-connectivity of a set-system is the minimum number of vertices that must be removed (along with any resulting empty edges) to obtain a non-connected set-system or singleton. Note that this means a single node has vertex-connectivity 0.

			Triangle begins:
    1
    2   0
    4   1   0
    9   3   2   0
   29  14  10   6   0
  209 157 128  91  54   0

Column k = 0 is A306505.
Column k = 1 is A261006 (clutters), if we assume A261006(0) = A261006(1) = 0.
Column k = 2 is A305028 (blobs), if we assume A305028(0) = A305028(2) = 0.
Except for the first column, same as A327358 (the covering case).
The labeled version is A327806.
Cf. A014466, A293606, A326704, A326950, A327062, A327334, A327350, A327351, A327805, A327808.

cp's OEIS Frontend

A327352 Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of antichains of nonempty subsets of {1..n} with spanning edge-connectivity k.

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A327060 Number of non-isomorphic weight-n weak antichains of multisets where every two vertices appear together in some edge (cointersecting).

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A327356 Number of connected separable antichains of nonempty sets covering n vertices (vertex-connectivity 1).

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A327806 Triangle read by rows where T(n,k) is the number of antichains of sets with n vertices and vertex-connectivity >= k.

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A327059 Number of pairwise intersecting set-systems covering a subset of {1..n} whose dual is a weak antichain.

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A327424 Number of unlabeled, non-connected or empty antichains of nonempty subsets of {1..n}.

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A327438 Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of unlabeled antichains of nonempty subsets of {1..n} with spanning edge-connectivity k.

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A327808 Number of unlabeled, disconnected, nonempty antichains of subsets of {1..n}.

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A327807 Triangle read by rows where T(n,k) is the number of unlabeled antichains of sets with n vertices and vertex-connectivity >= k.

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