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 A355517 Number of nonisomorphic systems enumerated by A334254; that is, the number of inequivalent closure operators on a set of n elements where all singletons are closed. 1
 1, 2, 1, 4, 50, 7443, 95239971 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS The T_1 axiom states that all singleton sets {x} are closed. For n>1, this property implies strictness (meaning that the empty set is closed). LINKS Table of n, a(n) for n=0..6. Dmitry I. Ignatov, Supporting iPython code for counting nonequivalent closure systems w.r.t. the T_1 separation axiom, Github repository Eric Weisstein's World of Mathematics, Separation Axioms Wikipedia, Separation Axiom EXAMPLE a(0) = 1 counts the empty set, while a(1) = 2 counts {{1}} and {{},{1}}. For a(2) = 1 the closure system is as follows: {{1,2},{1},{2},{}}. The a(3) = 4 inequivalent set-systems of closed sets are: {{1,2,3},{1},{2},{3},{}} {{1,2,3},{1,2},{1},{2},{3},{}} {{1,2,3},{1,2},{1,3},{1},{2},{3},{}} {{1,2,3},{1,2},{1,3},{2,3},{1},{2},{3},{}}. CROSSREFS The number of all closure operators is given in A102896, while A193674 contains the number of all nonisomorphic ones. For T_1 closure operators and their strict counterparts, see A334254 and A334255, respectively; the only difference is a(1). Cf. A326960, A326961, A326979. Sequence in context: A061655 A009830 A053374 * A227050 A093876 A322334 Adjacent sequences: A355514 A355515 A355516 * A355518 A355519 A355520 KEYWORD nonn,hard,more AUTHOR Dmitry I. Ignatov, Jul 05 2022 STATUS approved

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Last modified June 2 14:24 EDT 2023. Contains 363097 sequences. (Running on oeis4.)