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%I #23 Feb 04 2024 12:38:36
%S 1,1,1,4,167,165791
%N Number of strict closure operators on a set of n elements such that all pairs of distinct points can be separated by clopen sets.
%C A closure operator is strict if the empty set is closed.
%C Two distinct points x,y in X are separated by a set H if x is an element of H and y is not an element of H.
%C Also the number of S_2 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 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>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Closure_operator">Closure operator</a>
%e The a(3) = 4 set-systems of closed sets:
%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 Subsets[Range[n], {2}],
%t MemberQ[F,
%t _?(H |-> With[{H1 = Complement[Range[n], H]},
%t MemberQ[F, H1] && MemberQ[H, #[[1]]
%t ] && MemberQ[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, A358152, A356544.
%K nonn,hard,more
%O 0,4
%A _Tian Vlasic_, Oct 31 2022
%E a(5) from _Christian Sievers_, Feb 04 2024
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