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
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}}
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.
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Table[Length[Select[Subsets[Subsets[Range[n]]],SubsetQ[#,Union@@@Tuples[#,2]]&&SubsetQ[#,Intersection@@@Tuples[#,2]]&]],{n,0,3}]
A326905
BII-numbers of set-systems (without {}) closed under intersection.
Original entry on oeis.org
0, 1, 2, 4, 5, 6, 8, 16, 17, 21, 24, 32, 34, 38, 40, 56, 64, 65, 66, 68, 69, 70, 72, 80, 81, 85, 88, 96, 98, 102, 104, 120, 128, 256, 257, 261, 273, 277, 321, 325, 337, 341, 384, 512, 514, 518, 546, 550, 578, 582, 610, 614, 640, 896, 1024, 1025, 1026, 1028
Offset: 1
The sequence of all set-systems closed under intersection together with their BII-numbers begins:
0: {}
1: {{1}}
2: {{2}}
4: {{1,2}}
5: {{1},{1,2}}
6: {{2},{1,2}}
8: {{3}}
16: {{1,3}}
17: {{1},{1,3}}
21: {{1},{1,2},{1,3}}
24: {{3},{1,3}}
32: {{2,3}}
34: {{2},{2,3}}
38: {{2},{1,2},{2,3}}
40: {{3},{2,3}}
56: {{3},{1,3},{2,3}}
64: {{1,2,3}}
65: {{1},{1,2,3}}
66: {{2},{1,2,3}}
68: {{1,2},{1,2,3}}
The case with union instead of intersection is
A326875.
The case closed under union and intersection is
A326913.
Set-systems closed under intersection and containing the vertex set are
A326903.
Set-systems closed under intersection are
A326901, with unlabeled version
A326904.
-
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[0,100],SubsetQ[bpe/@bpe[#],Intersection@@@Tuples[bpe/@bpe[#],2]]&]
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
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.
-
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 *)
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