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.

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A319762 Number of non-isomorphic intersecting set multipartitions (multisets of sets) of weight n with empty intersection.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 1, 4, 9, 24
Offset: 0

Views

Author

Gus Wiseman, Sep 27 2018

Keywords

Comments

A set multipartition is intersecting if no two parts are disjoint. The weight of a set multipartition 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(6) = 1 through a(9) = 9 set multipartitions:
6: {{1,2},{1,3},{2,3}}
7: {{1,3},{1,4},{2,3,4}}
8: {{1,2},{1,3,4},{2,3,4}}
   {{1,4},{1,5},{2,3,4,5}}
   {{2,4},{1,2,5},{3,4,5}}
   {{1,2},{1,3},{2,3},{2,3}}
9: {{1,3},{1,4,5},{2,3,4,5}}
   {{1,5},{1,6},{2,3,4,5,6}}
   {{2,5},{1,2,6},{3,4,5,6}}
   {{1,2,3},{2,4,5},{3,4,5}}
   {{1,3,5},{2,3,6},{4,5,6}}
   {{1,2},{1,3},{1,4},{2,3,4}}
   {{1,2},{1,3},{2,3},{1,2,3}}
   {{1,3},{1,4},{1,4},{2,3,4}}
   {{1,3},{1,4},{3,4},{2,3,4}}

Crossrefs

Cf. A007716, A049311, A281116, A283877, A305854, A306006, A316980, A317757, A318715, A318717.
Cf. A319752, A319759, A319763, A319764, A319765, A319779, A319781.

A319763 Number of non-isomorphic strict intersecting multiset partitions (sets of multisets) of weight n with empty intersection.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 2, 12, 46, 181
Offset: 0

Views

Author

Gus Wiseman, Sep 27 2018

Keywords

Comments

A multiset partition is intersecting if no two parts are disjoint. 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(6) = 1 through a(8) = 12 multiset partitions:
6: {{1,2},{1,3},{2,3}}
7: {{1,2},{1,3},{2,3,3}}
   {{1,3},{1,4},{2,3,4}}
8: {{1,2},{1,3},{2,2,3,3}}
   {{1,2},{1,3},{2,3,3,3}}
   {{1,2},{1,3},{2,3,4,4}}
   {{1,2},{1,3,3},{2,3,3}}
   {{1,2},{1,3,4},{2,3,4}}
   {{1,3},{1,4},{2,3,4,4}}
   {{1,3},{1,1,2},{2,3,3}}
   {{1,3},{1,2,2},{2,3,3}}
   {{1,4},{1,5},{2,3,4,5}}
   {{2,3},{1,2,4},{3,4,4}}
   {{2,4},{1,2,3},{3,4,4}}
   {{2,4},{1,2,5},{3,4,5}}

Crossrefs

Cf. A007716, A281116, A283877, A305854, A306006, A316980, A317757, A318715, A318717.
Cf. A319752, A319759, A319762, A319764, A319765, A319779, A319781.

A319764 Number of non-isomorphic intersecting set systems of weight n with empty intersection.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 1, 3, 8, 18
Offset: 0

Views

Author

Gus Wiseman, Sep 27 2018

Keywords

Comments

A set system is a finite set of finite nonempty sets. It is intersecting if no two parts are disjoint. The weight of a set system 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(6) = 1 through a(9) = 8 set systems:
6: {{1,2},{1,3},{2,3}}
7: {{1,3},{1,4},{2,3,4}}
8: {{1,2},{1,3,4},{2,3,4}}
   {{1,4},{1,5},{2,3,4,5}}
   {{2,4},{1,2,5},{3,4,5}}
9: {{1,3},{1,4,5},{2,3,4,5}}
   {{1,5},{1,6},{2,3,4,5,6}}
   {{2,5},{1,2,6},{3,4,5,6}}
   {{1,2,3},{2,4,5},{3,4,5}}
   {{1,3,5},{2,3,6},{4,5,6}}
   {{1,2},{1,3},{1,4},{2,3,4}}
   {{1,2},{1,3},{2,3},{1,2,3}}
   {{1,3},{1,4},{3,4},{2,3,4}}

Crossrefs

Cf. A007716, A281116, A283877, A305854, A306006, A316980, A317752, A317755, A317757, A318715, A318717.
Cf. A319752, A319759, A319762, A319763, A319765, A319779, A319781.

A319782 Number of non-isomorphic intersecting strict T_0 multiset partitions of weight n.

Original entry on oeis.org

1, 1, 1, 4, 7, 17, 42, 98, 248, 631, 1657
Offset: 0

Views

Author

Gus Wiseman, Sep 27 2018

Keywords

Comments

A multiset partition is intersecting iff no two parts are disjoint. The weight of a multiset partition is the sum of sizes of its parts. 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 T_0 condition means the dual is strict.

Examples

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

Crossrefs

Cf. A007716, A049311, A059201, A281116, A283877, A305843, A305854, A306006, A316980.
Cf. A319755, A319759, A319760, A319765, A319779, A319787, A319789.

A305855 Number of unlabeled spanning intersecting antichains on n vertices.

Original entry on oeis.org

1, 1, 1, 3, 9, 72, 3441, 47170585
Offset: 0

Views

Author

Gus Wiseman, Jun 11 2018

Keywords

Comments

An intersecting antichain S is a finite set of finite nonempty sets (edges), any two of which have a nonempty intersection, and none of which is a subset of any other. S is spanning if every vertex is contained in some edge.

Examples

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

Crossrefs

Cf. A001206, A006126, A051185, A261006, A283877, A304998, A305843, A305844, A305854-A305857.

Formula

a(n) = A305857(n) - A305857(n-1) for n > 0. - Andrew Howroyd, Aug 13 2019

Extensions

a(6) from Andrew Howroyd, Aug 13 2019
a(7) from Brendan McKay, May 11 2020

A305856 Number of unlabeled intersecting set-systems on up to n vertices.

Original entry on oeis.org

1, 2, 4, 14, 124, 14992, 1289845584
Offset: 0

Views

Author

Gus Wiseman, Jun 11 2018

Keywords

Comments

An intersecting set-system is a finite set of finite nonempty sets (edges), any two of which have a nonempty intersection.

Examples

			Non-isomorphic representatives of the a(3) = 14 intersecting set-systems:
  {}
  {{1}}
  {{1,2}}
  {{1,2,3}}
  {{2},{1,2}}
  {{3},{1,2,3}}
  {{1,3},{2,3}}
  {{2,3},{1,2,3}}
  {{3},{1,3},{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. A001206, A006126, A051185, A261006, A283877, A304982, A304996, A304998, A305843, A305844, A305854-A305857.

Extensions

a(5) from Andrew Howroyd, Aug 12 2019
a(6) from Bert Dobbelaere, Apr 28 2025

A305935 Number of labeled spanning intersecting set-systems on n vertices with no singletons.

Original entry on oeis.org

1, 0, 1, 12, 809, 1146800, 899927167353, 291136684655893185321964, 14704020783497694096988185391720223222562121969, 12553242487939982849962414795232892198542733492886483991398790450208264017757788101836749760
Offset: 0

Views

Author

Gus Wiseman, Jun 15 2018

Keywords

Comments

An intersecting set-system S is a finite set of finite nonempty sets (edges), any two of which have a nonempty intersection. S is spanning if every vertex is contained in some edge. A singleton is an edge containing only one vertex.

Examples

			The a(3) = 12 spanning intersecting set-systems with no singletons:
{{1,2,3}}
{{1,2},{1,3}}
{{1,2},{2,3}}
{{1,3},{2,3}}
{{1,2},{1,2,3}}
{{1,3},{1,2,3}}
{{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}}
{{1,2},{1,3},{2,3},{1,2,3}}

Crossrefs

Cf. A001206, A006126, A051185, A048143, A058891, A305001, A305843, A305844, A305854-A305857, A305999, A306000, A306001, A306008.

Formula

a(n) = A305843(n) - n * A003465(n-1).
Inverse binomial transform of A306000. - Andrew Howroyd, Aug 12 2019

Extensions

a(6)-a(8) from Giovanni Resta, Jun 20 2018
a(9) from Andrew Howroyd, Aug 12 2019

A306000 Number of labeled intersecting set-systems with no singletons covering some subset of {1,...,n}.

Original entry on oeis.org

1, 1, 2, 16, 864, 1150976, 899934060544, 291136684662192699604992, 14704020783497694096990514485197495566069661696, 12553242487939982849962414795232892198542733625222671042878037323112413463887484853594095616
Offset: 0

Views

Author

Gus Wiseman, Jun 16 2018

Keywords

Comments

An intersecting set-system S is a finite set of finite nonempty sets (edges), any two of which have a nonempty intersection. A singleton is an edge containing only one vertex.

Examples

			The a(3) = 16 set-systems:
  {}
  {{1,2}}
  {{1,3}}
  {{2,3}}
  {{1,2,3}}
  {{1,2},{1,3}}
  {{1,2},{2,3}}
  {{1,3},{2,3}}
  {{1,2},{1,2,3}}
  {{1,3},{1,2,3}}
  {{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}}
  {{1,2},{1,3},{2,3},{1,2,3}}

Crossrefs

Cf. A001206, A006126, A051185, A048143, A058891, A305001, A305843, A305844, A305854-A305857, A305935, A305999, A306001.

Formula

a(n) = A051185(n) - n*2^(2^(n-1)-1). - Andrew Howroyd, Aug 12 2019

Extensions

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

A306001 Number of unlabeled intersecting set-systems with no singletons on up to n vertices.

Original entry on oeis.org

1, 1, 2, 8, 84, 13000
Offset: 0

Views

Author

Gus Wiseman, Jun 16 2018

Keywords

Comments

An intersecting set-system S is a finite set of finite nonempty sets (edges), any two of which have a nonempty intersection. A singleton is an edge containing only one vertex.

Examples

			Non-isomorphic representatives of the a(3) = 8 set-systems:
{}
{{1,2}}
{{1,2,3}}
{{1,3},{2,3}}
{{2,3},{1,2,3}}
{{1,2},{1,3},{2,3}}
{{1,3},{2,3},{1,2,3}}
{{1,2},{1,3},{2,3},{1,2,3}}

Crossrefs

Cf. A001206, A051185, A048143, A261006, A058891, A261005, A304998, A305854-A305857, A305935, A305999, A306000.

Formula

a(n) = A305856(n) - A000612(n). - Andrew Howroyd, Aug 12 2019

Extensions

a(5) from Andrew Howroyd, Aug 12 2019

A319784 Number of non-isomorphic intersecting T_0 set systems of weight n.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 3, 5, 7, 14, 25
Offset: 0

Views

Author

Gus Wiseman, Sep 27 2018

Keywords

Comments

A multiset partition is intersecting if no two parts are disjoint. The weight of a multiset partition is the sum of sizes of its parts. 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 T_0 condition means the dual is strict.

Examples

			Non-isomorphic representatives of the a(1) = 1 through a(8) = 7 multiset partitions:
1: {{1}}
3: {{2},{1,2}}
4: {{1,3},{2,3}}
5: {{3},{1,3},{2,3}}
6: {{3},{2,3},{1,2,3}}
   {{1,2},{1,3},{2,3}}
   {{1,4},{2,4},{3,4}}
7: {{4},{1,3,4},{2,3,4}}
   {{1,3},{1,4},{2,3,4}}
   {{1,3},{2,3},{1,2,3}}
   {{1,4},{3,4},{2,3,4}}
   {{4},{1,4},{2,4},{3,4}}
8: {{1,5},{2,4,5},{3,4,5}}
   {{2,4},{3,4},{1,2,3,4}}
   {{2,4},{1,2,5},{3,4,5}}
   {{2,4},{1,3,4},{2,3,4}}
   {{3},{1,3},{2,3},{1,2,3}}
   {{4},{1,4},{3,4},{2,3,4}}
   {{1,5},{2,5},{3,5},{4,5}}

Crossrefs

Cf. A007716, A049311, A283877, A305843, A305854, A306006, A316980, A317752.
Cf. A319755, A319759, A319760, A319765, A319779, A319787, A319782, A319789.
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