|
%I #16 Apr 24 2020 01:02:13
%S 1,1,3,35,2039,1352390,75945052607,14087646108883940225
%N Number of strict closure operators on a set of n elements which satisfy the T_0 separation axiom.
%C The T_0 axiom states that the closure of {x} and {y} are different for distinct x and y.
%C A closure operator is strict if the empty set is closed.
%H R. S. R. Myers, J. Adámek, S. Milius, and H. Urbat, <a href="https://doi.org/10.1016/j.tcs.2015.03.035">Coalgebraic constructions of canonical nondeterministic automata</a>, Theoretical Computer Science, 604 (2015), 81-101.
%H B. Venkateswarlu and U. M. Swamy, <a href="https://doi.org/10.1134/S1995080218090329">T_0-Closure Operators and Pre-Orders</a>, Lobachevskii Journal of Mathematics, 39 (2018), 1446-1452.
%F a(n) = Sum_{k=0..n} Stirling1(n,k) * A102894(k). - _Andrew Howroyd_, Apr 20 2020
%e The a(0) = 1 through a(2) = 3 set-systems of closed sets:
%e {{}} {{1},{}} {{1,2},{1},{}}
%e {{1,2},{2},{}}
%e {{1,2},{1},{2},{}}
%Y The number of all strict closure operators is given in A102894.
%Y For all T0 closure operators, see A334252.
%Y For strict T1 closure operators, see A334255.
%Y A strict closure operator which preserves unions is called topological, see A001035.
%Y Cf. A326943, A326944, A326945.
%K nonn,more
%O 0,3
%A _Joshua Moerman_, Apr 20 2020
%E a(6)-a(7) from _Andrew Howroyd_, Apr 20 2020
|
