A358152 - OEIS

%I #31 Jul 20 2024 08:10:25

%S 1,1,2,8,121,18460,159273237

%N Number of strict closure operators on a set of n elements such that every point and every closed 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

%E a(5)-a(6) from _Christian Sievers_, Jul 20 2024