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%I #24 May 14 2024 04:15:37
%S 1,1,2,8,121
%N Number of strict closure operators on a set of n elements such that every point and every set not containing that point can be separated by clopen sets.
%C A closure operator is strict if the empty set is closed.
%C A point p in X and a subset A of X not containing p are separated by a set H if p is an element of H and A is a subset of X\H.
%C Also the number of S_3 convexities on a set of n elements in the sense of Chepoi.
%D G. M. Bergman, "Lattices, Closure Operators, and Galois Connections", pp. 173-212 in "An Invitation to General Algebra and Universal Constructions", Springer, (2015).
%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>
%e The a(3) = 8 set-systems of closed sets:
%e {{}, {1, 2, 3}}
%e {{}, {1}, {2, 3}, {1, 2, 3}}
%e {{}, {2}, {1, 3},{1, 2, 3}}
%e {{}, {3}, {1, 2}, {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[F_, n_] := AllTrue[
%t Flatten[(x |-> ({x, #} & /@ Select[F, FreeQ[#, x] &])) /@ Range[n],
%t 1], MemberQ[F,
%t _?(H |-> With[{H1 = Complement[Range[n], H]},
%t MemberQ[F, H1] && MemberQ[H, #[[1]]
%t ] && SubsetQ[H1, #[[2]]
%t ]])]&];
%t Table[Length@Select[Select[
%t Subsets[Subsets[Range[n]]],
%t And[
%t MemberQ[#, {}],
%t MemberQ[#, Range[n]],
%t SubsetQ[#, Intersection @@@ Tuples[#, 2]]] &
%t ], SeparatedPairQ[#, n] &], {n, 0, 4}]
%Y Cf. A334255, A358144, A356544.
%K nonn,hard,more
%O 0,3
%A _Tian Vlasic_, Nov 01 2022
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