The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A358152 Number of strict closure operators on a set of n elements such that every point and every set not containing that point can be separated by clopen sets. 4
 1, 1, 2, 8, 121 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS A closure operator is strict if the empty set is closed. 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. Also the number of S_3 convexities on a set of n elements in the sense of Chepoi. REFERENCES G. M. Bergman, "Lattices, Closure Operators, and Galois Connections", pp. 173-212 in "An Invitation to General Algebra and Universal Constructions", Springer, (2015). LINKS Table of n, a(n) for n=0..4. Victor Chepoi, Separation of Two Convex Sets in Convexity Structures EXAMPLE The a(3) = 8 set-systems of closed sets: {{}, {1, 2, 3}} {{}, {1}, {2, 3}, {1, 2, 3}} {{}, {2}, {1, 3},{1, 2, 3}} {{}, {3}, {1, 2}, {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[F_, n_] := AllTrue[ Flatten[(x |-> ({x, #} & /@ Select[F, FreeQ[#, x] &])) /@ Range[n], 1], MemberQ[F, _?(H |-> With[{H1 = Complement[Range[n], H]}, MemberQ[F, H1] && MemberQ[H, #[[1]] ] && SubsetQ[H1, #[[2]] ]])]&]; Table[Length@Select[Select[ Subsets[Subsets[Range[n]]], And[ MemberQ[#, {}], MemberQ[#, Range[n]], SubsetQ[#, Intersection @@@ Tuples[#, 2]]] & ], SeparatedPairQ[#, n] &], {n, 0, 4}] CROSSREFS Cf. A334255, A358144, A356544. Sequence in context: A112094 A009658 A147794 * A027530 A228064 A271846 Adjacent sequences: A358149 A358150 A358151 * A358153 A358154 A358155 KEYWORD nonn,hard,more,changed AUTHOR Tian Vlasic, Nov 01 2022 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified May 20 11:55 EDT 2024. Contains 372712 sequences. (Running on oeis4.)