A358144 - OEIS

%I #26 Jun 13 2024 11:36:17

%S 1,1,1,4,167,165791,19194240969

%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

%E a(6) from _Christian Sievers_, Jun 13 2024