A326967 -id:A326967 - cp's OEIS frontend

A326965 Number of set-systems on n vertices where every covered vertex is the unique common element of some subset of the edges.

1, 2, 5, 46, 19181, 2010327182, 9219217424630040409, 170141181796805106025395618012972506978, 57896044618658097536026644159052312978532934306727333157337631572314050272137

Offset: 0

Gus Wiseman, Aug 10 2019

nonn

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-system where no edge is a subset of any other. This sequence counts set-systems whose dual is a (strict) antichain, also called T_1 set-systems.

			The a(0) = 1 through a(2) = 5 set-systems:
  {}  {}     {}
      {{1}}  {{1}}
             {{2}}
             {{1},{2}}
             {{1},{2},{1,2}}

Set-systems are A058891.
T_0 set-systems are A326940.
The covering case is A326961.
The version with empty edges allowed is A326967.
Set-systems whose dual is a weak antichain are A326968.
The unlabeled version is A326972.
The BII_numbers of these set-systems are A326979.
Cf. A059052, A326951, A326966, A326970, A326971, A326976, A326977.

tmQ[eds_]:=Union@@Select[Intersection@@@Rest[Subsets[eds]],Length[#]==1&]==Union@@eds;
Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],tmQ]],{n,0,3}]

Binomial transform of A326961.

a(n) = A326967(n)/2.

A326951 Number of unlabeled sets of subsets of {1..n} where every covered vertex is the unique common element of some subset of the edges.

Original entry on oeis.org

2, 4, 8, 40, 2464

Offset: 0

Gus Wiseman, Aug 13 2019

Alternatively, these are unlabeled sets of subsets of {1..n} whose dual is a (strict) antichain, also called T_1 sets of subsets. The dual of a set of subsets has, for each vertex, one edge consisting of the indices (or positions) of the edges containing that vertex. An antichain is a set of subsets where no edge is a subset of any other.

			Non-isomorphic representatives of the a(0) = 2 through a(2) = 8 sets of subsets:
  {}    {}        {}
  {{}}  {{}}      {{}}
        {{1}}     {{1}}
        {{},{1}}  {{},{1}}
                  {{1},{2}}
                  {{},{1},{2}}
                  {{1},{2},{1,2}}
                  {{},{1},{2},{1,2}}

Unlabeled sets of subsets are A003180.
Unlabeled T_0 sets of subsets are A326949.
The labeled version is A326967.
The case without empty edges is A326972.
The covering case is A327011 (first differences).
Cf. A003181, A059052, A326960, A326965, A326969, A326974, A326976.

a(n) = 2 * A326972(n).

a(n) = Sum_{k = 0..n} A327011(k).

A326960 Number of sets of subsets of {1..n} covering all n vertices whose dual is a (strict) antichain, also called covering T_1 sets of subsets.

Original entry on oeis.org

2, 2, 4, 72, 38040, 4020463392, 18438434825136728352, 340282363593610211921722192165556850240, 115792089237316195072053288318104625954343609704705784618785209431974668731584

Offset: 0

Gus Wiseman, Aug 13 2019

nonn

Same as A059052 except with a(1) = 2 instead of 4.
The dual of a set of subsets 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 subsets where no edge is a subset of any other.
Alternatively, these are sets of subsets of {1..n} covering all n vertices where every vertex is the unique common element of some subset of the edges.

			The a(0) = 2 through a(2) = 4 sets of subsets:
  {}    {{1}}     {{1},{2}}
  {{}}  {{},{1}}  {{},{1},{2}}
                  {{1},{2},{1,2}}
                  {{},{1},{2},{1,2}}

Covering sets of subsets are A000371.
Covering T_0 sets of subsets are A326939.
The case without empty edges is A326961.
The non-covering version is A326967.
Cf. A003181, A059052, A059523, A319639, A326951, A326965, A326974, A326976, A326977, A326979.

Table[Length[Select[Subsets[Subsets[Range[n]]],Length[Union[Select[Intersection@@@Rest[Subsets[#]],Length[#]==1&]]]==n&]],{n,0,3}]

Binomial transform of A326967.

A326969 Number of sets of subsets of {1..n} whose dual is a weak antichain.

Original entry on oeis.org

2, 4, 12, 112, 38892

Offset: 0

Gus Wiseman, Aug 10 2019

The dual of a set of subsets 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) = 2 through a(2) = 12 sets of subsets:
  {}    {}        {}
  {{}}  {{}}      {{}}
        {{1}}     {{1}}
        {{},{1}}  {{2}}
                  {{1,2}}
                  {{},{1}}
                  {{},{2}}
                  {{1},{2}}
                  {{},{1,2}}
                  {{},{1},{2}}
                  {{1},{2},{1,2}}
                  {{},{1},{2},{1,2}}

Sets of subsets whose dual is strict are A326941.
The BII-numbers of set-systems whose dual is a weak antichain are A326966.
Sets of subsets whose dual is a (strict) antichain are A326967.
The case without empty edges is A326968.
Cf. A001146, A059052, A326951, A326970, A326971, A326975, A326978.

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]]],stableQ[dual[#],SubsetQ]&]],{n,0,3}]

a(n) = 2 * A326968(n).

a(n) = 2 * Sum_{k = 0..n} binomial(n, k) * A326970(k).

A327017 Number of non-isomorphic multiset partitions of weight n where every vertex, as a multiset of weight 1, is the multiset-meet of some subset of the edges.

Original entry on oeis.org

1, 1, 2, 4, 9, 19, 49, 115, 310, 830, 2383

Offset: 0

Gus Wiseman, Aug 15 2019

The multiset-meet of a collection of multisets has as underlying set the intersection of their underlying sets and as multiplicities the minima of their multiplicities.

			Non-isomorphic representatives of the a(1) = 1 through a(5) = 19 multiset partitions:
    {1}  {1}{1}  {1}{11}    {1}{111}      {1}{1111}
         {1}{2}  {1}{1}{1}  {1}{1}{11}    {1}{1}{111}
                 {1}{2}{2}  {1}{2}{12}    {1}{11}{11}
                 {1}{2}{3}  {1}{2}{22}    {1}{12}{22}
                            {1}{1}{1}{1}  {1}{2}{122}
                            {1}{1}{2}{2}  {1}{2}{222}
                            {1}{2}{2}{2}  {1}{1}{1}{11}
                            {1}{2}{3}{3}  {1}{1}{2}{22}
                            {1}{2}{3}{4}  {1}{2}{2}{12}
                                          {1}{2}{2}{22}
                                          {1}{2}{3}{23}
                                          {1}{2}{3}{33}
                                          {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}
                                          {1}{2}{3}{4}{5}

Cf. A007716, A059523, A326961, A326965, A326967, A326972, A326974, A326976, A326977, A326979, A327012.

cp's OEIS Frontend

A326965 Number of set-systems on n vertices where every covered vertex is the unique common element of some subset of the edges.

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A326951 Number of unlabeled sets of subsets of {1..n} where every covered vertex is the unique common element of some subset of the edges.

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A326960 Number of sets of subsets of {1..n} covering all n vertices whose dual is a (strict) antichain, also called covering T_1 sets of subsets.

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A326969 Number of sets of subsets of {1..n} whose dual is a weak antichain.

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A327017 Number of non-isomorphic multiset partitions of weight n where every vertex, as a multiset of weight 1, is the multiset-meet of some subset of the edges.

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