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 1-9 of 9 results.

A318129 Number of sets of nonempty subsets of {1,...,n} with intersection {}.

Original entry on oeis.org

1, 1, 3, 91, 31827, 2147158387, 9223372011085950171, 170141183460469231602560095290109272523, 57896044618658097711785492504343953923912733397452774312538303978325772978595
Offset: 0

Views

Author

Gus Wiseman, Aug 18 2018

Keywords

Examples

			The a(2) = 3 sets of sets are {}, {{1},{2}}, {{1},{2},{1,2}}.

Crossrefs

Cf. A000371, A001146, A003465,A040996, A119563, A131288.
Cf. A318128, A318130, A318131, A318132.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Rest[Subsets[Range[n]]]],Or[#=={},Intersection@@#=={}]&]],{n,0,4}]

Formula

Binomial transform of A318128.
a(n) = A318130(n) - 2^(2^n - 1). [corrected]

A318131 Number of non-isomorphic sets of finite (possibly empty) sets with union {1,2,...,n} and intersection {}.

Original entry on oeis.org

1, 1, 6, 60, 3836, 37325360, 25626412263611792, 67516342973185974276922865448446208, 2871827610052485009904013737758920847534777143951264797898686184985092096
Offset: 0

Views

Author

Gus Wiseman, Aug 18 2018

Keywords

Examples

			Non-isomorphic representatives of the a(2) = 6 sets of sets:
  {{1},{2}}
  {{},{1,2}}
  {{},{1},{2}}
  {{},{1},{1,2}}
  {{1},{2},{1,2}}
  {{},{1},{2},{1,2}}

Links

Crossrefs

Cf. A000371, A000612, A003465, A055621, A119563, A131288, A283877, A293606, A304997.
Cf. A318128, A318129, A318130, A318132.

Programs

  • Mathematica
    sysnorm[m_]:=If[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[Length[Union[sysnorm/@Select[Subsets[Subsets[Range[n]]],And[Union@@#===Range[n],Intersection@@#=={}]&]]],{n,4}]

Formula

a(n) = 2*(A055621(n) - A055621(n-1)) = 2*(A000612(n) - 2*A000612(n-1) + A000612(n-2)) for n >= 2. - Andrew Howroyd, Jan 29 2024

Extensions

a(5) onwards from Andrew Howroyd, Jan 29 2024

A318132 Number of non-isomorphic set-systems (finite sets of finite nonempty sets) with union {1,2,...,n} and intersection {}.

Original entry on oeis.org

1, 0, 2, 26, 1884, 18660728, 12813206113141264, 33758171486592987125648226573752576, 1435913805026242504952006868879460423733630400489039411798068453617852416
Offset: 0

Views

Author

Gus Wiseman, Aug 18 2018

Keywords

Examples

			Non-isomorphic representatives of the a(3) = 26 set-systems:
  {{1},{2,3}}
  {{1},{2},{3}}
  {{1},{2},{1,3}}
  {{1},{2},{1,2,3}}
  {{1},{1,2},{2,3}}
  {{1},{2,3},{1,2,3}}
  {{1,2},{1,3},{2,3}}
  {{1},{2},{3},{1,2}}
  {{1},{2},{3},{1,2,3}}
  {{1},{2},{1,2},{1,3}}
  {{1},{2},{1,3},{2,3}}
  {{1},{2},{1,2},{1,2,3}}
  {{1},{2},{1,3},{1,2,3}}
  {{1},{1,2},{1,3},{2,3}}
  {{1},{1,2},{2,3},{1,2,3}}
  {{1},{2},{3},{1,2},{1,3}}
  {{1,2},{1,3},{2,3},{1,2,3}}
  {{1},{2},{3},{1,2},{1,2,3}}
  {{1},{2},{1,2},{1,3},{2,3}}
  {{1},{2},{1,2},{1,3},{1,2,3}}
  {{1},{2},{1,3},{2,3},{1,2,3}}
  {{1},{1,2},{1,3},{2,3},{1,2,3}}
  {{1},{2},{3},{1,2},{1,3},{2,3}}
  {{1},{2},{3},{1,2},{1,3},{1,2,3}}
  {{1},{2},{1,2},{1,3},{2,3},{1,2,3}}
  {{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}

Links

Crossrefs

Cf. A000371, A000612, A003465, A055621, A119563, A131288, A283877, A293606, A304997.
Cf. A318128, A318129, A318130, A318131.

Programs

  • Mathematica
    sysnorm[m_]:=If[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[Length[Union[sysnorm/@Select[Subsets[Rest[Subsets[Range[n]]]],And[Union@@#===Range[n],Intersection@@#=={}]&]]],{n,4}]

Formula

a(n) = A055621(n) - 2*A055621(n-1) = A000612(n) - 3*A000612(n-1) + 2*A000612(n-2) for n >= 2. - Andrew Howroyd, Jan 29 2024

Extensions

a(5) onwards from Andrew Howroyd, Jan 29 2024

A326365 Number of intersecting antichains 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, 1, 23, 1834, 1367903, 229745722873, 423295077919493525420
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 an antichain if no part is a subset of any other, and is intersecting if no two parts are disjoint.

Examples

			The a(4) = 23 intersecting antichains with empty intersection:
  {{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

Intersecting antichain covers are A305844.
Intersecting covers with empty intersection are A326364.
Antichain covers with empty intersection are A305001.
The binomial transform is the non-covering case A326366.
Covering, intersecting antichains with empty intersection are A326365.
Cf. A006126, A007363, A014466, A051185, A058891, A305843, A307249, A318128, 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}],Or[Intersection[#1,#2]=={},SubsetQ[#1,#2]]&],And[Union@@#==Range[n],#=={}||Intersection@@#=={}]&]],{n,0,4}]

Extensions

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

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

A318130 Number of sets of subsets of {1,...,n} with intersection {}.

Original entry on oeis.org

2, 3, 11, 219, 64595, 4294642035, 18446744047940725979, 340282366920938463334247399005993378251, 115792089237316195423570985008687907850547725730273056332267095982282337798563
Offset: 0

Views

Author

Gus Wiseman, Aug 18 2018

Keywords

Examples

			The a(2) = 11 sets of sets:
  {}
  {{}}
  {{},{1}}
  {{},{2}}
  {{1},{2}}
  {{},{1,2}}
  {{},{1},{2}}
  {{},{1},{1,2}}
  {{},{2},{1,2}}
  {{1},{2},{1,2}}
  {{},{1},{2},{1,2}}

Crossrefs

Cf. A000371, A001146, A003465, A119563, A131288, A318128, A318129.
Cf. A318128, A318129, A318131, A318132.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Subsets[Range[n]]],Or[#=={},Intersection@@#=={}]&]],{n,0,4}]

Formula

Binomial transform of A131288.
Inverse binomial transform of A119563(n) = 2^(2^n) + 2^n - 1.

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

A326373 Number of intersecting set systems with empty intersection (meaning there is no vertex in common to all the edges) on n vertices.

Original entry on oeis.org

1, 1, 1, 3, 435, 989555, 887050136795, 291072121058024908202443, 14704019422368226413236661148207899662350666147, 12553242487939461785560846872353486129110194529637343578112251094358919036718815137721635299
Offset: 0

Views

Author

Gus Wiseman, Jul 01 2019

Keywords

Comments

A set system (set of sets) is intersecting if no two edges are disjoint.

Examples

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

Crossrefs

The inverse binomial transform is the covering case A326364.
Set systems with empty intersection are A318129.
Intersecting set systems are A051185.
Intersecting antichains with empty intersection are A326366.
Cf. A000371, A006126, A007363, A014466, A058891, A305844, A307249, A318128, A326361, A326362, A326363, A326365.

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[#=={}||Intersection@@#=={}]&]],{n,0,4}]

Formula

a(n) = A051185(n) - 1 - Sum_{k=1..n-1} binomial(n,k)*A000371(k). - 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 1-9 of 9 results.

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