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

%I #10 Dec 30 2019 10:06:45

%S 2,2,7,45,500,9053,257151,11161244,725343385,69407094565,

%T 9639771895398,1919182252611715,541764452276876719,

%U 214777343584048313318,118575323291814379721651,90492591258634595795504697,94844885130660856889237907260,135738086271526574073701454370969,263921383510041055422284977248713291

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

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

%F a(n) = A000798(n) + A006058(n). - _Jean-François Alcover_, Dec 30 2019, after _Gus Wiseman_'s comment in A006058.

%e The a(0) = 2 through a(2) = 7 sets of subsets:

%e {} {{1}} {{1,2}}

%e {{}} {{},{1}} {{},{1,2}}

%e {{1},{1,2}}

%e {{2},{1,2}}

%e {{},{1},{1,2}}

%e {{},{2},{1,2}}

%e {{},{1},{2},{1,2}}

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

%t (* Second program: *)

%t A000798 = Cases[Import["https://oeis.org/A000798/b000798.txt", "Table"], {_, _}][[All, 2]];

%t A006058 = Cases[Import["https://oeis.org/A006058/b006058.txt", "Table"], {_, _}][[All, 2]];

%t a[n_] := A006058[[n + 1]] + A000798[[n + 1]];

%t a /@ Range[0, 18] (* _Jean-François Alcover_, Dec 30 2019 *)

%Y Covering sets of subsets are A000371.

%Y The case without empty sets is A108798.

%Y The case with a single covering edge is A326878.

%Y The unlabeled version is A326898 for n > 0.

%Y The case closed only under union is A326906.

%Y The case closed only under intersection is (also) A326906.

%Y Cf. A000798, A001930, A003465, A006058, A306445, A326876, A326882, A326907, A326908.

%K nonn

%O 0,1

%A _Gus Wiseman_, Aug 04 2019

%E a(18) from A000798+A006058 by _Jean-François Alcover_, Dec 30 2019