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 A334253 Number of strict closure operators on a set of n elements which satisfy the T_0 separation axiom. 2
 1, 1, 3, 35, 2039, 1352390, 75945052607, 14087646108883940225 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS The T_0 axiom states that the closure of {x} and {y} are different for distinct x and y. A closure operator is strict if the empty set is closed. LINKS R. S. R. Myers, J. Adámek, S. Milius, and H. Urbat, Coalgebraic constructions of canonical nondeterministic automata, Theoretical Computer Science, 604 (2015), 81-101. B. Venkateswarlu and U. M. Swamy, T_0-Closure Operators and Pre-Orders, Lobachevskii Journal of Mathematics, 39 (2018), 1446-1452. FORMULA a(n) = Sum_{k=0..n} Stirling1(n,k) * A102894(k). - Andrew Howroyd, Apr 20 2020 EXAMPLE The a(0) = 1 through a(2) = 3 set-systems of closed sets: {{}}  {{1},{}}  {{1,2},{1},{}}                 {{1,2},{2},{}}                 {{1,2},{1},{2},{}} CROSSREFS The number of all strict closure operators is given in A102894. For all T0 closure operators, see A334252. For strict T1 closure operators, see A334255. A strict closure operator which preserves unions is called topological, see A001035. Cf. A326943, A326944, A326945. Sequence in context: A320845 A012499 A125530 * A068726 A263512 A093583 Adjacent sequences:  A334250 A334251 A334252 * A334254 A334255 A334256 KEYWORD nonn,more AUTHOR Joshua Moerman, Apr 20 2020 EXTENSIONS a(6)-a(7) from Andrew Howroyd, Apr 20 2020 STATUS approved

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Last modified September 18 06:40 EDT 2020. Contains 337166 sequences. (Running on oeis4.)