cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A319765 Number of non-isomorphic intersecting multiset partitions of weight n whose dual is also an intersecting multiset partition.

Original entry on oeis.org

1, 1, 3, 6, 15, 31, 74, 156, 358, 792, 1821
Offset: 0

Views

Author

Gus Wiseman, Sep 27 2018

Keywords

Comments

The dual of a multiset partition has, for each vertex, one part consisting of the indices (or positions) of the parts containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}.
A multiset partition is intersecting iff no two parts are disjoint. The dual of a multiset partition is intersecting iff every pair of distinct vertices appear together in some part.
The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

Examples

			Non-isomorphic representatives of the a(1) = 1 through a(4) = 15 multiset partitions:
1: {{1}}
2: {{1,1}}
   {{1,2}}
   {{1},{1}}
3: {{1,1,1}}
   {{1,2,2}}
   {{1,2,3}}
   {{1},{1,1}}
   {{2},{1,2}}
   {{1},{1},{1}}
4: {{1,1,1,1}}
   {{1,1,2,2}}
   {{1,2,2,2}}
   {{1,2,3,3}}
   {{1,2,3,4}}
   {{1},{1,1,1}}
   {{1},{1,2,2}}
   {{2},{1,2,2}}
   {{3},{1,2,3}}
   {{1,1},{1,1}}
   {{1,2},{1,2}}
   {{1,2},{2,2}}
   {{1},{1},{1,1}}
   {{2},{2},{1,2}}
   {{1},{1},{1},{1}}

Crossrefs

Cf. A007716, A281116, A283877, A305854, A306006, A316980, A316983, A317757, A319616.
Cf. A319752, A319766, A319767, A319768, A319769, A319773, A319774.

A319774 Number of intersecting set systems spanning n vertices whose dual is also an intersecting set system.

Original entry on oeis.org

1, 1, 2, 14, 814, 1174774, 909125058112, 291200434263385001951232
Offset: 0

Views

Author

Gus Wiseman, Sep 27 2018

Keywords

Comments

The dual of a multiset partition has, for each vertex, one part consisting of the indices (or positions) of the parts containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}.
A multiset partition is intersecting iff no two parts are disjoint. The dual of a multiset partition is intersecting iff every pair of distinct vertices appear together in some part.

Examples

			The a(3) = 14 set systems:
   {{1},{1,2},{1,2,3}}
   {{1},{1,3},{1,2,3}}
   {{2},{1,2},{1,2,3}}
   {{2},{2,3},{1,2,3}}
   {{3},{1,3},{1,2,3}}
   {{3},{2,3},{1,2,3}}
   {{1,2},{1,3},{2,3}}
   {{1,2},{1,3},{1,2,3}}
   {{1,2},{2,3},{1,2,3}}
   {{1,3},{2,3},{1,2,3}}
   {{1},{1,2},{1,3},{1,2,3}}
   {{2},{1,2},{2,3},{1,2,3}}
   {{3},{1,3},{2,3},{1,2,3}}
   {{1,2},{1,3},{2,3},{1,2,3}}

Crossrefs

Cf. A007716, A281116, A283877, A305854, A306006, A316980, A316983, A317757, A319616.
Cf. A319752, A319765, A319766, A319767, A319768, A319769.
Intersecting set-systems are A051185.
The unlabeled multiset partition version is A319773.
The covering case is A327037.
The version without strict dual is A327038.
Cointersecting set-systems are A327039.
The BII-numbers of these set-systems are A327061.
Cf. A003465, A305843, A305844, A326854, A327020, A327040, A327052.

Programs

  • Mathematica
    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}]],Union@@#==Range[n]&&UnsameQ@@dual[#]&&stableQ[#,Intersection[#1,#2]=={}&]&&stableQ[dual[#],Intersection[#1,#2]=={}&]&]],{n,0,3}] (* Gus Wiseman, Aug 19 2019 *)

Extensions

a(6)-a(7) from Christian Sievers, Aug 18 2024

A319767 Number of non-isomorphic intersecting set systems spanning n vertices whose dual is also an intersecting set system.

Original entry on oeis.org

1, 1, 1, 5, 73
Offset: 0

Views

Author

Gus Wiseman, Sep 27 2018

Keywords

Comments

The dual of a multiset partition has, for each vertex, one part consisting of the indices (or positions) of the parts containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}.
A multiset partition is intersecting iff no two parts are disjoint. The dual of a multiset partition is intersecting iff every pair of distinct vertices appear together in some part.
The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

Examples

			Non-isomorphic representatives of the a(1) = 1 through a(3) = 5 multiset partitions:
1: {{1}}
2: {{2},{1,2}}
3: {{3},{2,3},{1,2,3}}
   {{1,2},{1,3},{2,3}}
   {{1,3},{2,3},{1,2,3}}
   {{3},{1,3},{2,3},{1,2,3}}
   {{1,2},{1,3},{2,3},{1,2,3}}

Crossrefs

Cf. A007716, A281116, A283877, A305854, A306006, A316980, A316983, A317757, A319616.
Cf. A319752, A319765, A319766, A319768, A319769, A319773, A319774.

A319768 Number of non-isomorphic strict multiset partitions (sets of multisets) of weight n whose dual is a (not necessarily strict) intersecting multiset partition.

Original entry on oeis.org

1, 1, 2, 5, 11, 25, 63, 144, 364, 905, 2356
Offset: 0

Views

Author

Gus Wiseman, Sep 27 2018

Keywords

Comments

The dual of a multiset partition has, for each vertex, one part consisting of the indices (or positions) of the parts containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}.
The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.
A multiset partition is intersecting iff no two parts are disjoint. The dual of a multiset partition is intersecting iff every pair of distinct vertices appear together in some part.

Examples

			Non-isomorphic representatives of the a(1) = 1 through a(4) = 11 multiset partitions:
1: {{1}}
2: {{1,1}}
   {{1,2}}
3: {{1,1,1}}
   {{1,2,2}}
   {{1,2,3}}
   {{1},{1,1}}
   {{2},{1,2}}
4: {{1,1,1,1}}
   {{1,1,2,2}}
   {{1,2,2,2}}
   {{1,2,3,3}}
   {{1,2,3,4}}
   {{1},{1,1,1}}
   {{1},{1,2,2}}
   {{2},{1,2,2}}
   {{3},{1,2,3}}
   {{1,2},{2,2}}
   {{1},{2},{1,2}}

Crossrefs

Cf. A007716, A281116, A283877, A305854, A306006, A316980, A316983, A317757, A319616.
Cf. A319752, A319765, A319766, A319767, A319769, A319773, A319774.

A319769 Number of non-isomorphic intersecting set multipartitions (multisets of sets) of weight n whose dual is also an intersecting set multipartition.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 12, 16, 26, 38, 61
Offset: 0

Views

Author

Gus Wiseman, Sep 27 2018

Keywords

Comments

The dual of a multiset partition has, for each vertex, one part consisting of the indices (or positions) of the parts containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}.
The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.
A multiset partition is intersecting iff no two parts are disjoint. The dual of a multiset partition is intersecting iff every pair of distinct vertices appear together in some part.

Examples

			Non-isomorphic representatives of the a(1) = 1 through a(5) = 7 set multipartitions:
1: {{1}}
2: {{1,2}}
   {{1},{1}}
3: {{1,2,3}}
   {{2},{1,2}}
   {{1},{1},{1}}
4: {{1,2,3,4}}
   {{3},{1,2,3}}
   {{1,2},{1,2}}
   {{2},{2},{1,2}}
   {{1},{1},{1},{1}}
5: {{1,2,3,4,5}}
   {{4},{1,2,3,4}}
   {{2,3},{1,2,3}}
   {{2},{1,2},{1,2}}
   {{3},{3},{1,2,3}}
   {{2},{2},{2},{1,2}}
   {{1},{1},{1},{1},{1}}

Crossrefs

Cf. A007716, A281116, A283877, A305854, A306006, A316980, A316983, A317757, A319616.
Cf. A319752, A319765, A319766, A319767, A319768, A319773, A319774.

A319773 Number of non-isomorphic intersecting set systems of weight n whose dual is also an intersecting set system.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 2, 1, 2, 4, 5
Offset: 0

Views

Author

Gus Wiseman, Sep 27 2018

Keywords

Comments

The dual of a multiset partition has, for each vertex, one part consisting of the indices (or positions) of the parts containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}.
The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.
A multiset partition is intersecting iff no two parts are disjoint. The dual of a multiset partition is intersecting iff every pair of distinct vertices appear together in some part.

Examples

			Non-isomorphic representatives of the a(1) = 1 through a(10) = 5 set systems:
1:  {{1}}
3:  {{2},{1,2}}
6:  {{3},{2,3},{1,2,3}}
    {{1,2},{1,3},{2,3}}
7:  {{1,3},{2,3},{1,2,3}}
8:  {{2,4},{3,4},{1,2,3,4}}
    {{3},{1,3},{2,3},{1,2,3}}
9:  {{1,2,4},{1,3,4},{2,3,4}}
    {{4},{2,4},{3,4},{1,2,3,4}}
    {{1,2},{1,3},{1,4},{2,3,4}}
    {{1,2},{1,3},{2,3},{1,2,3}}
10: {{4},{3,4},{2,3,4},{1,2,3,4}}
    {{4},{1,2,4},{1,3,4},{2,3,4}}
    {{1,2},{2,4},{1,3,4},{2,3,4}}
    {{1,4},{2,4},{3,4},{1,2,3,4}}
    {{2,3},{2,4},{3,4},{1,2,3,4}}

Crossrefs

Cf. A007716, A281116, A283877, A305854, A306006, A316980, A316983, A317757, A319616.
Cf. A319752, A319765, A319766, A319767, A319768, A319769, A319774.
Showing 1-6 of 6 results.

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