cp's OEIS Frontend

A358144 Number of strict closure operators on a set of n elements such that all pairs of distinct points can be separated by clopen sets.

%I A358144 #26 Jun 13 2024 11:36:17
%S A358144 1,1,1,4,167,165791,19194240969
%N A358144 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 A358144 A closure operator is strict if the empty set is closed.
%C A358144 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 A358144 Also the number of S_2 convexities on a set of n elements in the sense of Chepoi.
%D A358144 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 A358144 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 A358144 Wikipedia, <a href="https://en.wikipedia.org/wiki/Closure_operator">Closure operator</a>
%e A358144 The a(3) = 4 set-systems of closed sets:
%e A358144   {{}, {1}, {2}, {3}, {1, 2}, {1, 3}, {1, 2, 3}}
%e A358144   {{}, {1}, {2}, {3}, {1, 2}, {2, 3}, {1, 2, 3}}
%e A358144   {{}, {1}, {2}, {3}, {1, 3}, {2, 3}, {1, 2, 3}}
%e A358144   {{}, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}
%t A358144 SeparatedPairQ[F_, n_] := AllTrue[
%t A358144 Subsets[Range[n], {2}],
%t A358144 MemberQ[F,
%t A358144 _?(H |-> With[{H1 = Complement[Range[n], H]},
%t A358144       MemberQ[F, H1] && MemberQ[H, #[[1]]
%t A358144 ] && MemberQ[H1, #[[2]]
%t A358144 ]])] &];
%t A358144 Table[Length@Select[Select[
%t A358144    Subsets[Subsets[Range[n]]],
%t A358144    And[
%t A358144      MemberQ[#, {}],
%t A358144      MemberQ[#, Range[n]],
%t A358144      SubsetQ[#, Intersection @@@ Tuples[#, 2]]] &
%t A358144    ], SeparatedPairQ[#, n] &] , {n, 0, 4}]
%Y A358144 Cf. A334255, A358152, A356544.
%K A358144 nonn,hard,more
%O A358144 0,4
%A A358144 _Tian Vlasic_, Oct 31 2022
%E A358144 a(5) from _Christian Sievers_, Feb 04 2024
%E A358144 a(6) from _Christian Sievers_, Jun 13 2024