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 1-3 of 3 results.

A326366 Number of intersecting antichains of nonempty subsets of {1..n} with empty intersection (meaning there is no vertex in common to all the edges).

Original entry on oeis.org

1, 1, 1, 2, 28, 1960, 1379273, 229755337549, 423295079757497714059
Offset: 0

Views

Author

Gus Wiseman, Jul 01 2019

Keywords

Comments

A set system (set of sets) is an antichain if no edge is a subset of any other, and is intersecting if no two edges are disjoint.

Examples

			The a(0) = 1 through a(4) = 28 intersecting antichains with empty intersection:
  {}  {}  {}  {}              {}
              {{12}{13}{23}}  {{12}{13}{23}}
                              {{12}{14}{24}}
                              {{13}{14}{34}}
                              {{23}{24}{34}}
                              {{12}{13}{234}}
                              {{12}{14}{234}}
                              {{12}{23}{134}}
                              {{12}{24}{134}}
                              {{13}{14}{234}}
                              {{13}{23}{124}}
                              {{13}{34}{124}}
                              {{14}{24}{123}}
                              {{14}{34}{123}}
                              {{23}{24}{134}}
                              {{23}{34}{124}}
                              {{24}{34}{123}}
                              {{12}{134}{234}}
                              {{13}{124}{234}}
                              {{14}{123}{234}}
                              {{23}{124}{134}}
                              {{24}{123}{134}}
                              {{34}{123}{124}}
                              {{12}{13}{14}{234}}
                              {{12}{23}{24}{134}}
                              {{13}{23}{34}{124}}
                              {{14}{24}{34}{123}}
                              {{123}{124}{134}{234}}

Crossrefs

The case with empty edges allowed is A326375.
Intersecting antichains of nonempty sets are A001206.
Intersecting set systems with empty intersection are A326373.
Antichains of nonempty sets with empty intersection are A006126 or A307249.
The inverse binomial transform is the covering case A326365.
Cf. A007363, A014466, A051185, A058891, A305001, A305843, A305844, A318128, A318129, A326361, A326362, A326363, A326364.

Programs

  • Mathematica
    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]]]];
Table[Length[Select[stableSets[Subsets[Range[n],{1,n}],Or[Intersection[#1,#2]=={},SubsetQ[#1,#2]]&],#=={}||Intersection@@#=={}&]],{n,0,4}]

Formula

a(n) = A326375(n) - 1.
a(n) = A001206(n+1) + A307249(n) - A014466(n). - Andrew Howroyd, Aug 14 2019

Extensions

a(7)-a(8) from Andrew Howroyd, Aug 14 2019

A326364 Number of intersecting set systems with empty intersection (meaning there is no vertex in common to all the edges) covering n vertices.

Original entry on oeis.org

1, 0, 0, 2, 426, 987404, 887044205940, 291072121051815578010398, 14704019422368226413234332571239460300433492086, 12553242487939461785560846872353486129110194397301168776798213375239447299205732561174066488
Offset: 0

Views

Author

Gus Wiseman, Jul 01 2019

Keywords

Comments

Covering means there are no isolated vertices. A set system (set of sets) is intersecting if no two edges are disjoint.

Examples

			The a(3) = 2 intersecting set systems with empty intersection:
  {{1,2},{1,3},{2,3}}
  {{1,2},{1,3},{2,3},{1,2,3}}

Crossrefs

Covering set systems with empty intersection are A318128.
Covering, intersecting set systems are A305843.
Covering, intersecting antichains with empty intersection are A326365.
Cf. A006126, A007363, A014466, A051185, A058891, A305844, A307249, A318129, A326361, A326362, A326363.

Programs

  • Mathematica
    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]]]];
Table[Length[Select[stableSets[Subsets[Range[n],{1,n}],Intersection[#1,#2]=={}&],And[Union@@#==Range[n],#=={}||Intersection@@#=={}]&]],{n,0,4}]

Formula

Inverse binomial transform of A326373. - Andrew Howroyd, Aug 12 2019

Extensions

a(6)-a(9) from Andrew Howroyd, Aug 12 2019

A326375 Number of intersecting antichains of subsets of {1..n} with empty intersection (meaning there is no vertex in common to all the edges).

Original entry on oeis.org

2, 2, 2, 3, 29, 1961, 1379274, 229755337550, 423295079757497714060
Offset: 0

Views

Author

Gus Wiseman, Jul 03 2019

Keywords

Comments

A set system (set of sets) is an antichain if no edge is a subset of any other, and is intersecting if no two edges are disjoint.

Examples

			The a(4) = 29 antichains:
  {}
  {{}}
  {{1,2},{1,3},{2,3}}
  {{1,2},{1,4},{2,4}}
  {{1,3},{1,4},{3,4}}
  {{2,3},{2,4},{3,4}}
  {{1,2},{1,3},{2,3,4}}
  {{1,2},{1,4},{2,3,4}}
  {{1,2},{2,3},{1,3,4}}
  {{1,2},{2,4},{1,3,4}}
  {{1,3},{1,4},{2,3,4}}
  {{1,3},{2,3},{1,2,4}}
  {{1,3},{3,4},{1,2,4}}
  {{1,4},{2,4},{1,2,3}}
  {{1,4},{3,4},{1,2,3}}
  {{2,3},{2,4},{1,3,4}}
  {{2,3},{3,4},{1,2,4}}
  {{2,4},{3,4},{1,2,3}}
  {{1,2},{1,3,4},{2,3,4}}
  {{1,3},{1,2,4},{2,3,4}}
  {{1,4},{1,2,3},{2,3,4}}
  {{2,3},{1,2,4},{1,3,4}}
  {{2,4},{1,2,3},{1,3,4}}
  {{3,4},{1,2,3},{1,2,4}}
  {{1,2},{1,3},{1,4},{2,3,4}}
  {{1,2},{2,3},{2,4},{1,3,4}}
  {{1,3},{2,3},{3,4},{1,2,4}}
  {{1,4},{2,4},{3,4},{1,2,3}}
  {{1,2,3},{1,2,4},{1,3,4},{2,3,4}}

Crossrefs

The case without empty edges is A326366.
Intersecting antichains are A326372.
Antichains of nonempty sets with empty intersection are A006126 or A307249.
Cf. A001206, A007363, A014466, A051185, A058891, A305001, A305843, A305844, A318128, A318129, A326363, A326365, A326373.

Programs

  • Mathematica
    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]]]];
Table[Length[Select[stableSets[Subsets[Range[n]],Or[Intersection[#1,#2]=={},SubsetQ[#1,#2]]&],#=={}||Intersection@@#=={}&]],{n,0,4}]

Formula

a(n) = A326366(n) + 1.

Extensions

a(7)-a(8) from Andrew Howroyd, Aug 14 2019
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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