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%I #51 Jun 13 2024 11:36:43
%S 0,1,4,35,857,84230,70711467
%N Number of strict closure operators on a set of n elements such that all pairs of nonempty disjoint closed sets can be separated by clopen sets.
%C A closure operator is strict if the empty set is closed.
%C Two nonempty disjoint subsets A and B of X are separated by a set H if A is a subset of H and B is a subset of X\H.
%C Also the number of S_4 (Kakutani separation property) convexities on a set of n elements in the sense of Chepoi.
%D G. M. Bergman. Lattices, Closure Operators, and Galois Connections. Springer, Cham. 2015. 173-212.
%H Victor Chepoi, <a href="https://www.researchgate.net/publication/2407147_Separation_Of_Two_Convex_Sets_In_Convexity_Structures">Separation of Two Convex Sets in Convexity Structures</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Closure_operator">Closure operator</a>
%e The a(3) = 35 set-systems of closed sets:
%e {{}, {1, 2, 3}}
%e {{}, {1}, {1, 2, 3}}
%e {{}, {2}, {1, 2, 3}}
%e {{}, {3}, {1, 2, 3}}
%e {{}, {1, 2}, {1, 2, 3}}
%e {{}, {1, 3}, {1, 2, 3}}
%e {{}, {2, 3}, {1, 2, 3}}
%e {{}, {1}, {1, 2}, {1, 2, 3}}
%e {{}, {1}, {1, 3}, {1, 2, 3}}
%e {{}, {1}, {2, 3}, {1, 2, 3}}
%e {{}, {2}, {1, 2}, {1, 2, 3}}
%e {{}, {2}, {1, 3}, {1, 2, 3}}
%e {{}, {2}, {2, 3}, {1, 2, 3}}
%e {{}, {3}, {1, 2}, {1, 2, 3}}
%e {{}, {3}, {1, 3}, {1, 2, 3}}
%e {{}, {3}, {2, 3}, {1, 2, 3}}
%e {{}, {1}, {2}, {1, 3}, {1, 2, 3}}
%e {{}, {1}, {2}, {2, 3}, {1, 2, 3}}
%e {{}, {1}, {3}, {1, 2}, {1, 2, 3}}
%e {{}, {1}, {3}, {2, 3}, {1, 2, 3}}
%e {{}, {1}, {1, 2}, {1, 3}, {1, 2, 3}}
%e {{}, {2}, {3}, {1, 2}, {1, 2, 3}}
%e {{}, {2}, {3}, {1, 3}, {1, 2, 3}}
%e {{}, {2}, {1, 2}, {2, 3}, {1, 2, 3}}
%e {{}, {3}, {1, 3}, {2, 3}, {1, 2, 3}}
%e {{}, {1}, {2}, {1, 2}, {1, 3}, {1, 2, 3}}
%e {{}, {1}, {2}, {1, 2}, {2, 3}, {1, 2, 3}}
%e {{}, {1}, {3}, {1, 2}, {1, 3}, {1, 2, 3}}
%e {{}, {1}, {3}, {1, 3}, {2, 3}, {1, 2, 3}}
%e {{}, {2}, {3}, {1, 2}, {2, 3}, {1, 2, 3}}
%e {{}, {2}, {3}, {1, 3}, {2, 3}, {1, 2, 3}}
%e {{}, {1}, {2}, {3}, {1, 2}, {1, 3}, {1, 2, 3}}
%e {{}, {1}, {2}, {3}, {1, 2}, {2, 3}, {1, 2, 3}}
%e {{}, {1}, {2}, {3}, {1, 3}, {2, 3}, {1, 2, 3}}
%e {{}, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}
%t SeparatedPairQ[A_][B_] := AnyTrue[A, And @@ MapThread[SubsetQ, {#, B}] &];
%t Table[Length[With[{X = Range[n]},
%t Select[Cases[Subsets@Subsets@X, {{}, ___, X}],
%t F |-> SubsetQ[F, Intersection @@@ Subsets[F, {2}]]
%t && AllTrue[Select[Subsets[Drop[F, 1], {2}], Apply[DisjointQ]], SeparatedPairQ[Select[{#, Complement[X, #]} & /@ F, MemberQ[F, #[[2]]] &]]]]]], {n, 0, 4}]
%Y Cf. A334255, A358144, A358152.
%K nonn,hard,more
%O 0,3
%A _Tian Vlasic_, Aug 11 2022
%E a(5)-a(6) from _Christian Sievers_, Jun 13 2024
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