A334253 - OEIS

%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