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 A334254 Number of closure operators on a set of n elements which satisfy the T_1 separation axiom. 5
 1, 2, 1, 8, 545, 702525, 66960965307 (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, On the Cryptomorphism between Davis' Subset Lattices, Atomic Lattices, and Closure Systems under T1 Separation Axiom, arXiv:2209.12256 [cs.DM], 2022. Dmitry I. Ignatov, Supporting iPython code for counting closure systems w.r.t. the T_1 separation axiom, Github repository Dmitry I. Ignatov, PDF of the supporting iPython notebook S. Mapes, Finite atomic lattices and resolutions of monomial ideals, J. Algebra, 379 (2013), 259-276. 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},{}} CROSSREFS The number of all closure operators is given in A102896. For T_0 closure operators, see A334252. For strict T_1 closure operators, see A334255, the only difference is a(1). Cf. A326960, A326961, A326979. Sequence in context: A013327 A359625 A009349 * A230582 A011186 A078088 Adjacent sequences: A334251 A334252 A334253 * A334255 A334256 A334257 KEYWORD nonn,more,hard AUTHOR Joshua Moerman, Apr 20 2020 EXTENSIONS a(6) from Dmitry I. Ignatov, Jul 03 2022 STATUS approved

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Last modified May 29 07:06 EDT 2024. Contains 372926 sequences. (Running on oeis4.)