login
A356544
Number of strict closure operators on a set of n elements such that all pairs of nonempty disjoint closed sets can be separated by clopen sets.
4
0, 1, 4, 35, 857, 84230, 70711467
OFFSET
0,3
COMMENTS
A closure operator is strict if the empty set is closed.
Two nonempty disjoint subsets A and B of X are separated by a set H if A is a subset of H and B is a subset of X\H.
Also the number of S_4 (Kakutani separation property) convexities on a set of n elements in the sense of Chepoi.
REFERENCES
G. M. Bergman. Lattices, Closure Operators, and Galois Connections. Springer, Cham. 2015. 173-212.
EXAMPLE
The a(3) = 35 set-systems of closed sets:
{{}, {1, 2, 3}}
{{}, {1}, {1, 2, 3}}
{{}, {2}, {1, 2, 3}}
{{}, {3}, {1, 2, 3}}
{{}, {1, 2}, {1, 2, 3}}
{{}, {1, 3}, {1, 2, 3}}
{{}, {2, 3}, {1, 2, 3}}
{{}, {1}, {1, 2}, {1, 2, 3}}
{{}, {1}, {1, 3}, {1, 2, 3}}
{{}, {1}, {2, 3}, {1, 2, 3}}
{{}, {2}, {1, 2}, {1, 2, 3}}
{{}, {2}, {1, 3}, {1, 2, 3}}
{{}, {2}, {2, 3}, {1, 2, 3}}
{{}, {3}, {1, 2}, {1, 2, 3}}
{{}, {3}, {1, 3}, {1, 2, 3}}
{{}, {3}, {2, 3}, {1, 2, 3}}
{{}, {1}, {2}, {1, 3}, {1, 2, 3}}
{{}, {1}, {2}, {2, 3}, {1, 2, 3}}
{{}, {1}, {3}, {1, 2}, {1, 2, 3}}
{{}, {1}, {3}, {2, 3}, {1, 2, 3}}
{{}, {1}, {1, 2}, {1, 3}, {1, 2, 3}}
{{}, {2}, {3}, {1, 2}, {1, 2, 3}}
{{}, {2}, {3}, {1, 3}, {1, 2, 3}}
{{}, {2}, {1, 2}, {2, 3}, {1, 2, 3}}
{{}, {3}, {1, 3}, {2, 3}, {1, 2, 3}}
{{}, {1}, {2}, {1, 2}, {1, 3}, {1, 2, 3}}
{{}, {1}, {2}, {1, 2}, {2, 3}, {1, 2, 3}}
{{}, {1}, {3}, {1, 2}, {1, 3}, {1, 2, 3}}
{{}, {1}, {3}, {1, 3}, {2, 3}, {1, 2, 3}}
{{}, {2}, {3}, {1, 2}, {2, 3}, {1, 2, 3}}
{{}, {2}, {3}, {1, 3}, {2, 3}, {1, 2, 3}}
{{}, {1}, {2}, {3}, {1, 2}, {1, 3}, {1, 2, 3}}
{{}, {1}, {2}, {3}, {1, 2}, {2, 3}, {1, 2, 3}}
{{}, {1}, {2}, {3}, {1, 3}, {2, 3}, {1, 2, 3}}
{{}, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}
MATHEMATICA
SeparatedPairQ[A_][B_] := AnyTrue[A, And @@ MapThread[SubsetQ, {#, B}] &];
Table[Length[With[{X = Range[n]},
Select[Cases[Subsets@Subsets@X, {{}, ___, X}],
F |-> SubsetQ[F, Intersection @@@ Subsets[F, {2}]]
&& AllTrue[Select[Subsets[Drop[F, 1], {2}], Apply[DisjointQ]], SeparatedPairQ[Select[{#, Complement[X, #]} & /@ F, MemberQ[F, #[[2]]] &]]]]]], {n, 0, 4}]
CROSSREFS
KEYWORD
nonn,hard,more
AUTHOR
Tian Vlasic, Aug 11 2022
EXTENSIONS
a(5)-a(6) from Christian Sievers, Jun 13 2024
STATUS
approved