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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A326906 Number of sets of subsets of {1..n} that are closed under union and cover all n vertices.

Original entry on oeis.org

2, 2, 8, 90, 4542, 2747402, 151930948472, 28175295407840207894
Offset: 0

Views

Author

Gus Wiseman, Aug 03 2019

Keywords

Comments

Differs from A102895 in having a(0) = 2 instead of 1.

Examples

			The a(0) = 2 through a(2) = 8 sets of subsets:
  {}    {{1}}     {{1,2}}
  {{}}  {{},{1}}  {{},{1,2}}
                  {{1},{1,2}}
                  {{2},{1,2}}
                  {{},{1},{1,2}}
                  {{},{2},{1,2}}
                  {{1},{2},{1,2}}
                  {{},{1},{2},{1,2}}

Crossrefs

The case without empty sets is A102894.
The case with a single covering edge is A102895.
Binomial transform is A102897.
The case also closed under intersection is A326878 for n > 0.
The same for intersection instead of union is (also) A326906.
The unlabeled version is A326907.
Cf. A000798, A102896, A102897, A108800, A193675, A306445, A326880, A326881, A326883.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Subsets[Range[n]]],Union@@#==Range[n]&&SubsetQ[#,Union@@@Tuples[#,2]]&]],{n,0,3}]

Formula

a(n) = 2 * A102894(n).

A326883 Number of unlabeled set-systems with {} that are closed under intersection and cover n vertices.

Original entry on oeis.org

1, 1, 4, 22, 302, 28630, 216533404, 5592325966377736
Offset: 0

Views

Author

Gus Wiseman, Jul 30 2019

Keywords

Examples

			Non-isomorphic representatives of the a(0) = 1 through a(3) = 22 set-systems:
  {{}}  {{}{1}}  {{}{12}}        {{}{123}}
                 {{}{1}{2}}      {{}{1}{23}}
                 {{}{2}{12}}     {{}{3}{123}}
                 {{}{1}{2}{12}}  {{}{1}{2}{3}}
                                 {{}{23}{123}}
                                 {{}{1}{3}{23}}
                                 {{}{2}{3}{123}}
                                 {{}{3}{13}{23}}
                                 {{}{1}{23}{123}}
                                 {{}{3}{23}{123}}
                                 {{}{1}{2}{3}{23}}
                                 {{}{1}{2}{3}{123}}
                                 {{}{2}{3}{13}{23}}
                                 {{}{1}{3}{23}{123}}
                                 {{}{2}{3}{23}{123}}
                                 {{}{3}{13}{23}{123}}
                                 {{}{1}{2}{3}{13}{23}}
                                 {{}{1}{2}{3}{23}{123}}
                                 {{}{2}{3}{13}{23}{123}}
                                 {{}{1}{2}{3}{12}{13}{23}}
                                 {{}{1}{2}{3}{13}{23}{123}}
                                 {{}{1}{2}{3}{12}{13}{23}{123}}

Crossrefs

The case also closed under union is A001930.
The connected case (i.e., with maximum) is A108798.
The same for union instead of intersection is (also) A108798.
The non-covering case is A108800.
The labeled case is A326881.
Cf. A000798, A102894, A102895, A102897, A193674, A193675, A306445, A326875, A326878, A326880.

Formula

a(n) = A108800(n) - A108800(n-1) for n > 0. - Andrew Howroyd, Aug 10 2019

Extensions

a(5)-a(7) from Andrew Howroyd, Aug 10 2019

A326898 Number of unlabeled topologies with up to n points.

Original entry on oeis.org

1, 2, 5, 14, 47, 186, 904, 5439, 41418, 404501, 5122188, 84623842, 1828876351, 51701216248, 1908493827243, 91755916071736, 5729050033597431
Offset: 0

Views

Author

Gus Wiseman, Aug 02 2019

Keywords

Examples

			Non-isomorphic representatives of the a(0) = 1 through a(3) = 14 topologies:
  {}  {}     {}            {}
      {}{1}  {}{1}         {}{1}
             {}{12}        {}{12}
             {}{2}{12}     {}{123}
             {}{1}{2}{12}  {}{2}{12}
                           {}{3}{123}
                           {}{23}{123}
                           {}{1}{2}{12}
                           {}{1}{23}{123}
                           {}{3}{23}{123}
                           {}{2}{3}{23}{123}
                           {}{3}{13}{23}{123}
                           {}{2}{3}{13}{23}{123}
                           {}{1}{2}{3}{12}{13}{23}{123}

Crossrefs

Partial sums of A001930.
The labeled version is A326878.
Cf. A000612, A000798, A003180, A108800, A193675, A306445, A326867, A326882.

A326904 Number of unlabeled set-systems (without {}) on n vertices that are closed under intersection.

Original entry on oeis.org

1, 2, 4, 10, 38, 368, 29328, 216591692, 5592326399531792
Offset: 0

Views

Author

Gus Wiseman, Aug 04 2019

Keywords

Comments

A set-system is a finite set of finite nonempty sets, so no two edges of such a set-system can be disjoint.
Apart from the offset the same as A193675. - R. J. Mathar, Aug 09 2019

Examples

			Non-isomorphic representatives of the a(0) = 1 through a(3) = 10 set-systems:
  {}  {}     {}           {}
      {{1}}  {{1}}        {{1}}
             {{1,2}}      {{1,2}}
             {{2},{1,2}}  {{1,2,3}}
                          {{2},{1,2}}
                          {{3},{1,2,3}}
                          {{2,3},{1,2,3}}
                          {{3},{1,3},{2,3}}
                          {{3},{2,3},{1,2,3}}
                          {{3},{1,3},{2,3},{1,2,3}}

Crossrefs

The covering case is A108800(n - 1).
The case with an edge containing all of the vertices is A193674(n - 1).
The case with union instead of intersection is A193674.
The labeled version is A326901.
Cf. A000798, A001930, A006058, A102895, A102898, A326876, A326866, A326878, A326882, A326903, A326906.

Formula

a(n > 0) = 2 * A193674(n - 1).

A326908 Number of non-isomorphic sets of subsets of {1..n} that are closed under union and intersection.

Original entry on oeis.org

2, 4, 9, 23, 70, 256, 1160, 6599, 48017, 452518, 5574706, 90198548, 1919074899, 53620291147, 1962114118390, 93718030190126, 5822768063787557
Offset: 0

Views

Author

Gus Wiseman, Aug 03 2019

Keywords

Examples

			Non-isomorphic representatives of the a(0) = 2 through a(3) = 23 sets of subsets:
  {}    {}       {}              {}
  {{}}  {{}}     {{}}            {{}}
        {{1}}    {{1}}           {{1}}
        {{}{1}}  {{12}}          {{12}}
                 {{}{1}}         {{}{1}}
                 {{}{12}}        {{123}}
                 {{2}{12}}       {{}{12}}
                 {{}{2}{12}}     {{}{123}}
                 {{}{1}{2}{12}}  {{2}{12}}
                                 {{3}{123}}
                                 {{}{2}{12}}
                                 {{23}{123}}
                                 {{}{3}{123}}
                                 {{}{23}{123}}
                                 {{}{1}{2}{12}}
                                 {{3}{23}{123}}
                                 {{}{1}{23}{123}}
                                 {{}{3}{23}{123}}
                                 {{3}{13}{23}{123}}
                                 {{}{2}{3}{23}{123}}
                                 {{}{3}{13}{23}{123}}
                                 {{}{2}{3}{13}{23}{123}}
                                 {{}{1}{2}{3}{12}{13}{23}{123}}

Crossrefs

The labeled version is A306445.
Taking first differences and prepending 1 gives A326898.
Taking second differences and prepending two 1's gives A001930.
Cf. A000612, A000798, A003180, A108798, A108800, A193675, A326867, A326876, A326878, A326882, A326883.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Subsets[Range[n]]],SubsetQ[#,Union@@@Tuples[#,2]]&&SubsetQ[#,Intersection@@@Tuples[#,2]]&]],{n,0,3}]
Previous Showing 11-15 of 15 results.

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