A108798 -id:A108798 - cp's OEIS frontend

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

1, 1, 4, 22, 302, 28630, 216533404, 5592325966377736

Offset: 0

Gus Wiseman, Jul 30 2019

			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}}

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.

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

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

A326907 Number of non-isomorphic sets of subsets of {1..n} that are closed under union and cover all n vertices. First differences of A193675.

Original entry on oeis.org

2, 2, 6, 28, 330, 28960, 216562364, 5592326182940100

Offset: 0

Gus Wiseman, Aug 03 2019

Differs from A108800 in having a(0) = 2 instead of 1.

			Non-isomorphic representatives of the a(0) = 2 through a(3) = 28 sets of sets:
  {}    {{1}}    {{12}}          {{123}}
  {{}}  {{}{1}}  {{}{12}}        {{}{123}}
                 {{2}{12}}       {{3}{123}}
                 {{}{2}{12}}     {{23}{123}}
                 {{1}{2}{12}}    {{}{3}{123}}
                 {{}{1}{2}{12}}  {{}{23}{123}}
                                 {{1}{23}{123}}
                                 {{3}{23}{123}}
                                 {{13}{23}{123}}
                                 {{}{1}{23}{123}}
                                 {{}{3}{23}{123}}
                                 {{}{13}{23}{123}}
                                 {{2}{3}{23}{123}}
                                 {{2}{13}{23}{123}}
                                 {{3}{13}{23}{123}}
                                 {{12}{13}{23}{123}}
                                 {{}{2}{3}{23}{123}}
                                 {{}{2}{13}{23}{123}}
                                 {{}{3}{13}{23}{123}}
                                 {{}{12}{13}{23}{123}}
                                 {{2}{3}{13}{23}{123}}
                                 {{3}{12}{13}{23}{123}}
                                 {{}{2}{3}{13}{23}{123}}
                                 {{}{3}{12}{13}{23}{123}}
                                 {{2}{3}{12}{13}{23}{123}}
                                 {{}{2}{3}{12}{13}{23}{123}}
                                 {{1}{2}{3}{12}{13}{23}{123}}
                                 {{}{1}{2}{3}{12}{13}{23}{123}}

The case without empty sets is A108798.
The case with a single covering edge is A108800.
First differences of A193675.
The case also closed under intersection is A326898 for n > 0.
The labeled version is A326906.
The same for union instead of intersection is (also) A326907.
Cf. A001930, A102895, A108798, A193674, A193675, A326880, A326881, A326883, A326898, A326908.

a(7) added from A108800 by Andrew Howroyd, Aug 10 2019

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

Gus Wiseman, Aug 03 2019

			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}}

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.

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

A326909 Number of sets of subsets of {1..n} closed under union and intersection and covering all of the vertices.

Original entry on oeis.org

2, 2, 7, 45, 500, 9053, 257151, 11161244, 725343385, 69407094565, 9639771895398, 1919182252611715, 541764452276876719, 214777343584048313318, 118575323291814379721651, 90492591258634595795504697, 94844885130660856889237907260, 135738086271526574073701454370969, 263921383510041055422284977248713291

Offset: 0

Gus Wiseman, Aug 04 2019

nonn

Differs from A326878 in having a(0) = 2 instead of 1.

			The a(0) = 2 through a(2) = 7 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}}

Covering sets of subsets are A000371.
The case without empty sets is A108798.
The case with a single covering edge is A326878.
The unlabeled version is A326898 for n > 0.
The case closed only under union is A326906.
The case closed only under intersection is (also) A326906.
Cf. A000798, A001930, A003465, A006058, A306445, A326876, A326882, A326907, A326908.

Table[Length[Select[Subsets[Subsets[Range[n]]],Union@@#==Range[n]&&SubsetQ[#,Union[Union@@@Tuples[#,2],Intersection@@@Tuples[#,2]]]&]],{n,0,3}]
(* Second program: *)
A000798 = Cases[Import["https://oeis.org/A000798/b000798.txt", "Table"], {, }][[All, 2]];
A006058 = Cases[Import["https://oeis.org/A006058/b006058.txt", "Table"], {, }][[All, 2]];
a[n_] := A006058[[n + 1]] + A000798[[n + 1]];
a /@ Range[0, 18] (* Jean-François Alcover, Dec 30 2019 *)

a(n) = A000798(n) + A006058(n). - Jean-François Alcover, Dec 30 2019, after Gus Wiseman's comment in A006058.

a(18) from A000798+A006058 by Jean-François Alcover, Dec 30 2019

A299116 The number of sparse union-closed sets. That is, the number of union-closed sets on n elements containing the empty set and the universe, such that in average each set (not counting the empty set) has at most n/2 elements.

Original entry on oeis.org

0, 0, 0, 2, 27, 3133, 5777931

Offset: 1

Gunnar Brinkmann, Feb 05 2018

If there is a counterexample to the union-closed set conjecture, it is a sparse union-closed set.

G. Brinkmann and R. Deklerck, Generation of Union-Closed Sets and Moore Families, arXiv:1701.03751 [math.CO], 2017.
G. Brinkmann and R. Deklerck, Generation of Union-Closed Sets and Moore Families, Journal of Integer Sequences, Vol.21 (2018), Article 18.1.7.

Cf. A108798, A102894, A193674, A102896.

cp's OEIS Frontend

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

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A326907 Number of non-isomorphic sets of subsets of {1..n} that are closed under union and cover all n vertices. First differences of A193675.

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

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A326909 Number of sets of subsets of {1..n} closed under union and intersection and covering all of the vertices.

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A299116 The number of sparse union-closed sets. That is, the number of union-closed sets on n elements containing the empty set and the universe, such that in average each set (not counting the empty set) has at most n/2 elements.

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