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 A334255 Number of strict closure operators on a set of n elements which satisfy the T_1 separation axiom. 7
 1, 1, 1, 8, 545, 702525, 66960965307 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS The T_1 axiom states that all singleton sets {x} are closed. A closure operator is strict if the empty set is closed. LINKS Table of n, a(n) for n=0..6. Dmitry I. Ignatov, Supporting iPython code for counting closure systems w.r.t. the T_1 separation axiom, Github repository Dmitry I. Ignatov, Supporting iPython notebook Eric Weisstein's World of Mathematics, Separation Axioms Wikipedia, Separation Axiom EXAMPLE The a(3) = 8 set-systems of closed sets: {{1,2,3},{1},{2},{3},{}} {{1,2,3},{1,2},{1},{2},{3},{}} {{1,2,3},{1,3},{1},{2},{3},{}} {{1,2,3},{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 Table[Length[ Select[Subsets[Subsets[Range[n]]], And[MemberQ[#, {}], MemberQ[#, Range[n]], SubsetQ[#, Intersection @@@ Tuples[#, 2]], SubsetQ[#, Map[{#} &, Range[n]]]] &]], {n, 0, 4}] (* Tian Vlasic, Jul 29 2022 *) CROSSREFS The number of all strict closure operators is given in A102894. For all strict T_0 closure operators, see A334253. For T_1 closure operators, see A334254. Cf. A326960, A326961, A326979. Sequence in context: A200706 A266207 A027536 * A174252 A181682 A248331 Adjacent sequences: A334252 A334253 A334254 * A334256 A334257 A334258 KEYWORD nonn,more AUTHOR Joshua Moerman, Apr 24 2020 EXTENSIONS a(6) from Dmitry I. Ignatov, Jul 03 2022 STATUS approved

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Last modified June 5 05:23 EDT 2023. Contains 363130 sequences. (Running on oeis4.)